5
$s = 2 + {1\over 2}\sqrt 2 = \Nn{2.70710678118654}$
Rigid.
Proved by Frits Göbel
in early 1979.
10
$s = 3 + {1\over 2}\sqrt 2 = \Nn{3.70710678118654}$
Found by Frits Göbel in early 1979.
Proved by Walter Stromquist in 2003.
Explore group
11
$s = {}^{8}🔒 = \Nn{3.87708359002281}$
$s^8 - 20s^7 + 178s^6 - 842s^5 + 1923s^4 - 496s^3 - 6754s^2 + 12420s - 6865 = 0$
Rigid.
Found by Walter Trump
in 1979.
17
$s = {}^{18}🔒 = \Nn{4.67553009360455}$
$4775s^{18}-190430s^{17}+3501307s^{16}-39318012s^{15}+300416928s^{14}-1640654808s^{13}+6502333062s^{12}-18310153596s^{11}+32970034584s^{10}-18522084588s^9-93528282146s^8+350268230564s^7-662986732745s^6+808819596154s^5-660388959899s^4+358189195800s^3-126167814419s^2+26662976550s-2631254953=0$
Found by John Bidwell
in 1998.
Based on packing found by Pertti Hämäläinen in 1980.
18
$s = {7\over 2} + {1\over 2}\sqrt 7 = \Nn{4.82287565553229}$
Found by Pertti Hämäläinen
in 1980.
Pictured alternative with minimal rotated squares found by Mats Gustafsson in 1981.
19
$s = 3 + {4\over 3}\sqrt 2 = \Nn{4.88561808316412}$
Found first by Robert Wainwright
in late 1979.
Based on packing found by Charles F. Cottingham in early 1979.
26
$s = {7\over 2} + {3\over 2}\sqrt 2 = \Nn{5.62132034355964}$
Found by Erich Friedman
in 1997.
Unextends the $s(37)$ found by Evert Stenlund in early 1980.
27
$s = 5 + {1\over 2}\sqrt 2 = \Nn{5.70710678118654}$
Found by Frits Göbel
in early 1979.
Explore group
28
$s = 3 + 2 \sqrt 2 = \Nn{5.82842712474619}$
Found by Frits Göbel
in early 1979.
Explore group
29
$s = \Nn{5.93434180499654}$
Found by Thierry Gensane and Philippe Ryckelynck in April 2004, using a computer program they wrote.
Fits an $s(n^2\!-\!n\!-\!1)$ pattern found
by Joe DeVincentis in April 2014.
Explore group
37
$s = {}^{8}🔒 = \Nn{6.59861960924436}$
$6s^4-(208+64\sqrt{2})s^3+(2058+850\sqrt{2})s^2-(7936+3658\sqrt{2})s+11163+5502\sqrt{2}=0$
$36s^8-2496s^7+59768s^6-733760s^5+5289248s^4-23462672s^3+63458276s^2-96673872s+64068561=0$
Found by David W. Cantrell in September 2002.
Improves upon the $s(37)$ found by Evert Stenlund in early 1980.
38
$s = 6 + {1\over 2}\sqrt 2 = \Nn{6.70710678118654}$
Found by Frits Göbel
in early 1979.
Explore group
39
$s = {}^{8}🔒 = \Nn{6.81880916998841}$
$5184s^8-197568s^7+3200144s^6-28651016s^5+154197141s^4-506760114s^3+981374688s^2-1005617394s+408278853=0$
Found by David W. Cantrell
in August 2002.
40
$s = 4 + 2 \sqrt 2 = \Nn{6.82842712474619}$
Rigid.
Found by Frits Göbel
in early 1979.
Explore group
41
$s = {}^{4}🔒 = \Nn{6.93786550630255}$
$s^4-16s^3+95s^2-218s-34=0$
Found by Joe DeVincentis
in April 2014.
Fits an $s(n^2\!-\!n\!-\!1)$ pattern.
Explore group
50
$s = {}^{8}🔒 = \Nn{7.59861960924436}$
$6s^4-(232+64\sqrt{2})s^3+(2718+1042\sqrt{2})s^2-(12700+5550\sqrt{2})s+21371+10074\sqrt{2}=0$
$36s^8-2784s^7+78248s^6-1146800s^5+9944448s^4-53242000s^3+173869324s^2-319180600s+253748689=0$
Found by David W. Cantrell
in September 2002, by adding
an "L" to the $s(37)$ he found.
Improves upon the $s(50)$ found by Evert Stenlund in early 1980.
51
$s = {}^{28}🔒 = \Nn{7.70435372947124}$
$36864s^{28}-6340608s^{27}+502050816s^{26}-24636665856s^{25}+847410746368s^{24}-21857473382400s^{23}+441080063406080s^{22}-7168010813250560s^{21}+95780257115813376s^{20}-1068807979173627904s^{19}+10079527432131681024s^{18}-81076261200222141184s^{17}+560144016315152943424s^{16}-3340596154679285521280s^{15}+17248154822575215485952s^{14}-77154000981112955287360s^{13}+298459379274993606556192s^{12}-993878287428748511469056s^{11}+2827560064086331516654992s^{10}-6798876862709272559608016s^9+13620557443692132080422196s^8-22318847802748398169997192s^7+29208515137302727559556744s^6-29572821499263810227200404s^5+22139938904533199326391397s^4-11407627350518593079154528s^3+3525180882798952592954446s^2-436679755165931930913236s-28766318325274882531199=0$
Found by Károly Hajba
in July 2009.
52
$s = 7 + {1\over 2}\sqrt 2 = \Nn{7.70710678118654}$
Found by Frits Göbel
in early 1979.
Explore group
53
$s = {}^{32}🔒 = \Nn{7.82290804413284}$
$2401s^{32}-470596s^{31}+43819034s^{30}-2575626200s^{29}+107064827921s^{28}-3339730095676s^{27}+80881055947292s^{26}-1549634326032740s^{25}+23649840659276976s^{24}-285549917423038976s^{23}+2644342639858467944s^{22}-16975793431501140604s^{21}+42160111960108077186s^{20}+604801559847940668640s^{19}-10323577086818326059762s^{18}+91423093604241671442960s^{17}-549498547429972204556388s^{16}+2140033946880564583409424s^{15}-2116245094452089256368748s^{14}-47398133328019683350564492s^{13}+513623762890535143202861016s^{12}-3539168668169055445821007956s^{11}+19554643118966991967193841680s^{10}-91052982556166904610991305520s^9+358000595889063667996962518449s^8-1171244223040008871491078909092s^7+3123815163273613989845145831648s^6-6628626979480975566515883381512s^5+10851380919900152459339530602076s^4-13122619975009229974564386454832s^3+10946326483876900501448971940288s^2-5556443722823912253342620368384s+1267690936215846749823290569552=0$
Found by David W. Cantrell
in September 2002.
Improved by David W. Cantrell
in December 2024.
54
$\begin{aligned}s &= 7-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{7.84666719284348}\end{aligned}$
Found by David W. Cantrell
in October 2005.
Improved by Joe DeVincentis
in April 2014.
55
$s = \Nn{7.95419161110664}$
Found by Joe DeVincentis in April 2014, using a computer program he wrote. Improved by David Ellsworth and David W. Cantrell in June & August 2023 and David Ellsworth in November 2024.
Fits an $s(n^2\!-\!n\!-\!1)$ pattern.
Explore group
65
$s = 5 + {5\over 2}\sqrt 2 = \Nn{8.53553390593273}$
Found by Frits Göbel
in early 1979.
Explore group
66
$s = 3 + 4 \sqrt 2 = \Nn{8.65685424949238}$
Found by Evert Stenlund in early 1980.
67
$s = 8 + {1\over 2}\sqrt 2 = \Nn{8.70710678118654}$
Found by Evert Stenlund
in early 1980, extending the $s(52)$
found by Frits Göbel in early 1979.
Explore group
68
$s = {13\over 3} + 2 \sqrt 5 = \Nn{8.80546928833291}
Found by Sigvart Brendberg
in June 2023.
69
$s = {}^{82}🔒 = \Nn{8.82721205592900}$
$52389094428262881s^{82}-28863139436366651460s^{81}+7840436786580754561842s^{80}-1399864630898909951672184s^{79}+184777024966383679131379203s^{78}-19229480097533386652981194668s^{77}+1643178003450476327369002864080s^{76}-118561352785653984081132853368864s^{75}+7372351836836707441183744339971015s^{74}-401254176764396680092337021141946484s^{73}+19350157008010415954432078062713291394s^{72}-834969623551779032213936610875479861512s^{71}+32500264420943843392373991413578392058093s^{70}-1148852629892528066579108553164478473663708s^{69}+37092466248098270905023679715303792737820304s^{68}-1099206042418214352026228628885408398048015000s^{67}+30025320958251433175557289720600502032769753340s^{66}-758792087058505752402362438963674625699826919880s^{65}+17799410748369850870306914205805242294037335637896s^{64}-388686829570450651667791276249653981802721222714056s^{63}+7922061683854685568474881816199072307645318622376904s^{62}-151058341641411022199807974673871019724902497364765552s^{61}+2700552785792713834768094889293145620129036537676224092s^{60}-45354522344129825814676420288826173471599912259984496632s^{59}+716878712470410740335863321139824820808423827153710652804s^{58}-10682567284888720343007934969631240418818071811270135320816s^{57}+150320784390672934545124162608853418121767017935787301000808s^{56}-2000572646236355172723818796429406996345627284039771483014960s^{55}+25219559033013693277083797294746787502373277261234753716013214s^{54}-301583920452466921147984771117351156297618201001262096981290160s^{53}+3426054385349213936246735756144263675017479361589187104582644952s^{52}-37026637210515130032012648558266141117178874708570301177143938096s^{51}+381221915869963598518466209504441332617716678547088629463788058492s^{50}-3744411601889467025365599805308355961072225438102130448464025588920s^{49}+35132927721859555152174976560750433704133787706160759119646097007600s^{48}-315304729246464792403348347852665200866883662496880642068223778829192s^{47}+2709948932058311309971179409319475857433539061059204130543172912232550s^{46}-22330292252239325190451014020603871952094854771701615927312762869972264s^{45}+176590827377409087722261448341442541695845580235159787543827412635270192s^{44}-1341409274447219282944440341226557560736584173610170808364041163179628592s^{43}+9794628200723929363909228342085371052888507149267241738990749330177446536s^{42}-68784984456991565723134237317800008579678347641360227701542986898597128624s^{41}+464787351026639375955250101748280481985644641368851374515808386357693042884s^{40}-3022573184259078701450135458529957385309345186803651626114551783083378175032s^{39}+18919089267049873225236080915564725310548628869115830052258169804904606332284s^{38}-113971460035925598073276330819280830203445312638283436481301177127978414813080s^{37}+660644243129473954993233623574173921633210380878554917654203983937559606764892s^{36}-3683382377823441838082957327165940185883796561462208003223994363396807558829680s^{35}+19741959358629662296400872197474154929765830845655143211973252362263291667066932s^{34}-101642479500862445314955859849362422289005748345703180795721307378605167190176216s^{33}+502223128747819353777858875489650546509178956355195996750910607953412777650946008s^{32}-2378848650747301593887480639497480434456215486100674152450449358032767827357494504s^{31}+10787427200018965953466228877088228967021423580833924967176436387993308084517771520s^{30}-46763657666979111364440110538290788706620901509781273614839323079790048897674112936s^{29}+193478197292846750318686197125966160724659499210376202873234094610563436453287357712s^{28}-762652613846301377253090541691216290609269954386886415663741638905969508874033669080s^{27}+2858810541382820711701247202901545177530055188476676694775559880120687456958481517369s^{26}-10170991995607092582144907594501215089484328088609684391404365725870234777077196212052s^{25}+34275552245333382898966081848057394622466895493701655338137448625314981168202736100870s^{24}-109180149865199065847120380278545633781677317125880517882096621147002464901674366232896s^{23}+328024240104468595897778174882791456088805151159650666727391352426213891813001741009597s^{22}-927474305514792318700089933609615567301057834087731929460292201817254239263791481291308s^{21}+2462182229902610406598305774365812710170450400050717006211185019057179425643813174021924s^{20}-6122044755330252945719750665463174625835643677192302695331089860574738738324123064544448s^{19}+14219984970544731850691516928796082391054234525105419850802193862381157708451060567522208s^{18}-30768851545907889218776308829677014260927583947632211969217096923859242673238301283472512s^{17}+61831404131179569857993652298053425544232611206764320680249716948926268397092525417269376s^{16}-115009817315259102016058959098678198023224949369106956091463566410698079424438533154919424s^{15}+197271311091301472792347653205833439690067290927728227908502214395626223917618164276099328s^{14}-310715226079337036201755817142663826462822486357621041087800464785427130883680776524049408s^{13}+447233873751878497967377512304289813779839139617400585441961124659571901207054515474723840s^{12}-584994487650569941937265070878539829783049696806373201094322271724763322754772388187897856s^{11}+690809670769485727048919721008636863534640513613632064766742686036251506800827979919523840s^{10}-730705950216779945965312115026670309649787853302475272108288646183442394632173792483868672s^9+685727465494402560587060223400049402456139486767982657035415974606333680206469376237371392s^8-564182795837916615774045743559109089033591178820776604035503312959295256878380021673099264s^7+400823651584041532933559377617252554932923674966442340917105411238495035002689607404879872s^6-241020915379745770711663822572215144075000506186967983082373034574012538132391946971250688s^5+119331539747892530196375157797097038574572404727228577993084411405965584791382011108392960s^4-46729898398085553837033675288544422050050908921747303951054359523991662277479822073528320s^3+13577207271788496430462938959054088460341797225685886738526659498529340720805243256832000s^2-2603186344462167626779756466825247201285474002427939337103647694067437869176377573376000s+247160402287431680471138762403368003391572385877539215982119721342810263983667281920000=0$
Found by Maurizio Morandi
in June 2010.
Improved by David W. Cantrell
in August 2023.
70
$s = {}^{4}🔒 = \Nn{8.88166675700900}$
$23s^4-742s^3+8848s^2-45876s+86229=0$
Found by Joe DeVincentis
in April 2014.
71
$s = {}^{4}🔒 = \Nn{8.96028765944389}$
$s^4-20s^3+151s^2-468s+12=0$
Found by Joe DeVincentis
in April 2014.
Fits an $s(n^2\!-\!n\!-\!1)$ pattern.
Explore group
82
$s = 6 + {5\over 2}\sqrt 2 = \Nn{9.53553390593273}$
Found by Frits Göbel in early 1979.
Adds two "L"s to $s(65)$.
83
$s = {}^{24}🔒 = \Nn{9.63482562092335}$
$46438209s^{24}+1718447880s^{23}-1304818741864s^{22}+154362940868008s^{21}-10223870917986092s^{20}+463012769729234068s^{19}-15608677475881443482s^{18}+410530364971106359132s^{17}-8675319117762080311978s^{16}+150196459602374087471728s^{15}-2158879193002672091253360s^{14}+25993038455067669296355532s^{13}-263613888105247221344935027s^{12}+2258335015809616506745502008s^{11}-16347943921555337654669478150s^{10}+99786776593815833271369617220s^9-511154425074511891757096094175s^8+2180187656593439512672814134216s^7-7652314463979073976449593048904s^6+21727853135387976484209118127392s^5-48671720700899577518293563957136s^4+82801528406446840092722047620736s^3-100540002112755895115349929950336s^2+77621257841393908308227797286912s-28634116465193128516311336597248=0$
Found by Károly Hajba in September 2024.
Improved upon the $s(83)$ found by Evert Stenlund in early 1980.
Improved by David W. Cantrell in November 2024.
Extends the $s(17)$ found by John Bidwell in 1998.
84
$s = 9 + {1\over 2}\sqrt 2 = \Nn{9.70710678118654}$
Found by Evert Stenlund
in early 1980, extending the $s(52)$
found by Frits Göbel in early 1979.
Explore group
85
$s = {11\over 2} + 3 \sqrt 2 = \Nn{9.74264068711928}$
Found by Erich Friedman
in 1997.
86
$s = {17\over 2} + {1\over 2}\sqrt 7 = \Nn{9.82287565553229}$
Found by Erich Friedman
in 1997.
Extends the alternative packing of the $s(18)$ found by Pertti Hämäläinen in 1980 found by Mats Gustafsson in 1981.
87
$s = {}^{23}🔒 = \Nn{9.83892657002494}$
$s^{23}-138s^{22}+8984s^{21}-366792s^{20}+10538108s^{19}-226899940s^{18}+3814912554s^{17}-51682985704s^{16}+579852353410s^{15}-5521708432172s^{14}+45434118479338s^{13}-324120092543232s^{12}+1970502666465045s^{11}-9840546468521178s^{10}+38505534857507358s^9-116163600689532188s^8+342445835317445719s^7-1749940540506500506s^6+11136111412137553730s^5-53096842537804640028s^4+169960636556199528165s^3-350586565237438644834s^2+425167991192928955284s-231741242909814395880=0$
Found by David W. Cantrell
in January 2025.
88
$s = {}^{20}🔒 = \Nn{9.88815305375857}$
$3528s^{10}-(300552+15456\sqrt{2})s^9+(11614660+1180832\sqrt{2})s^8-(268405824+40209136\sqrt{2})s^7+(4111948776+801750848\sqrt{2})s^6-(43682208312+10328732976\sqrt{2})s^5+(326223055436+89277369408\sqrt{2})s^4-(1692962073984+518553084040\sqrt{2})s^3+(5849274524474+1954912407552\sqrt{2})s^2-(12163170266098+4347871933856\sqrt{2})s+11572065260145+4353477802040\sqrt{2}=0$
$254016s^{20}-43279488s^{19}+3506260608s^{18}-179642577984s^{17}+6530192527760s^{16}-179088304328704s^{15}+3846118270819200s^{14}-66261902137415296s^{13}+930479746642904384s^{12}-10759858027891736896s^{11}+103070340120029179008s^{10}-819709665351861223904s^9+5405590814889373243192s^8-29412949608198679086720s^7+130831566348158107359392s^6-468664620024162429231904s^5+1321046745485882223459068s^4-2825402176181244872057384s^3+4315682289270565775115128s^2-4199847844458434080013540s+1959329723251932809573425=0$
Found by Erich Friedman in 1997, by extending the $s(41)$ found by Charles F. Cottingham in 1979.
Improved by David Ellsworth in November 2024, by adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Improved independently by both David W. Cantrell and David Ellsworth in January 2025.
This new technique, which will need to be applied to about 20 additional packings previously thought to be finished, independently found by both David W. Cantrell and David Ellsworth.
89
$s = 5 + {7\over 2}\sqrt 2 = \Nn{9.94974746830583}$
Found by Evert Stenlund in early 1980,
by continuing a pattern found by Frits Göbel in early 1979.
Explore group
101
$s = 7 + {5\over 2}\sqrt 2 = \Nn{10.53553390593273}$
Adds two "L"s to the $s(65)$ found
by Frits Göbel in early 1979.
102
$s = {}^{8}🔒 = \Nn{10.61138823373863}$
$24s^4-(1400+352\sqrt{2})s^3+(27061+10102\sqrt{2})s^2-(218629+97462\sqrt{2})s+641430+317240\sqrt{2}=0$
$576s^8-67200s^7+3011120s^6-72041376s^5+1033920257s^4-9243724322s^3+50697397293s^2-156795019420s+210150009700=0$
Found by Károly Hajba
in September 2024.
Extended the $s(37)$ found by Evert Stenlund in early 1980.
Improved by David W. Cantrell and
David Ellsworth in November 2024,
by extending the $s(37)$ found by
David W. Cantrell in September 2002.
Improved by David Ellsworth in December 2024.
Further improvement pending.
103, 104
$s = 10 + {1\over 2}\sqrt 2 = \Nn{10.70710678118654}$
Extends the $s(52)$ found by
Frits Göbel in early 1979.
Explore group
105
$s = {19\over 2} + {1\over 2}\sqrt 7 = \Nn{10.82287565553229}$
Adds an "L" to the $s(86)$ found by Erich Friedman in 1997.
106
$s = {}^{32}🔒 = \Nn{10.82297973416944}$
$2401s^{32}-701092s^{31}+96532058s^{30}-8317521660s^{29}+501462496833s^{28}-22376320364004s^{27}+760407348527794s^{26}-19848555869936524s^{25}+391901782318184024s^{24}-5471444559723346548s^{23}+39629249963743971218s^{22}+353696512578770314160s^{21}-16715802829777049603255s^{20}+292780863461637719269068s^{19}-3146950297418725382386108s^{18}+17996731060753457271434416s^{17}+58068494391477930003466013s^{16}-2710032149344351718373370304s^{15}+35897708426171881544444261010s^{14}-321777454480517334707593455472s^{13}+2318288875965343221612347387046s^{12}-15552954072951813301922897418028s^{11}+110324583017828739076547861980256s^{10}-814314195444111054964406531808040s^9+5535691601850017528577776913992458s^8-31551366481166818299554205010301876s^7+144126073330457054877480503221027356s^6-514883701839008702934553147517634876s^5+1404879821405123399570947930851054697s^4-2828245713344639081326540531975430356s^3+3961424449372054137804580730689208222s^2-3448216591537867586914544066365388952s+1404226509074988020217588819457924033=0$
Found by David Ellsworth
in November 2024, based on the
$s(53)$ found by David W. Cantrell in September 2002.
Improved by David W. Cantrell
in December 2024.
107
$\begin{aligned}s &= 10-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{10.84666719284348}\end{aligned}$
Found by Károly Hajba
in November 2024.
Is an alternative packing for an extension of the $s(54)$ found by Joe DeVincentis in April 2014.
108
$s = {}^{144}🔒 = \Nn{10.92591939016138}$
$827055562171106304s^{144}-1028746038533781676032s^{143}+614161678485829473038592s^{142}-236434676228380170030685824s^{141}+66980317656649750618706949945s^{140}-15177024487481458274595256127868s^{139}+2918547306910189530506516307375692s^{138}-495729497699611401035280356277680196s^{137}+76089742722234909946196429571333895452s^{136}-10671270893794158877895712177505617523828s^{135}+1375524471618065793880112212598599154653012s^{134}-163680131457973276468171788500327444221444244s^{133}+18041122005484577974071987969780608673747943250s^{132}-1845193374227011301793607038733620442135422735440s^{131}+175195858247481457503205356072668634514707875635464s^{130}-15447131106769150898435521166852265757689347117679048s^{129}+1266589004665859543076277348204652895182867619945255898s^{128}-96872971664861242811412554825190666410176955662558075472s^{127}+6939847565941400486198515625279425362463969118244043805152s^{126}-467586251641955795650527642523133741785627313804139361021412s^{125}+29712916686346155911816329503684280580564307321782343390920945s^{124}-1781869546071794107416303074386560117711940471499978105624204512s^{123}+100692141845407597467752420973765886037444159069768871392617305440s^{122}-5345881769519666404549787030353246578976419001556791199058788111868s^{121}+265745153241256297998761893367366891336827254195720886433343716673314s^{120}-12331900216784265711374361173174853992773142985602439082869045995043620s^{119}+533124642048160878797787516816371559269658695957365696289239653586152536s^{118}-21454346119467121162152752804366671634925654995122131901651080854842653908s^{117}+804137509033612105675830501719617753487476429965727315635094765154608429509s^{116}-28127107800477454224608783588306926887609791011395988349691834833987158467700s^{115}+921246416930684576450726414356413649291858434685374246183966749784918747685898s^{114}-28390357852514647575635304153251006733509079472468682793162434720668520497597508s^{113}+828166533055942389980323072853474773741844516945352000111380951692338661814562750s^{112}-23019859946745245564619050854509564382765776816954463496160143457200568875655263500s^{111}+613584341652374269884917793377224034449208943707706722281248832851205212434542935116s^{110}-15759013359756201601425954086600815916125169227043330927990424168462555090123951648000s^{109}+390943459868126235255883761901077907242607987729337591414340253957672403607173215398650s^{108}-9366575279423493930590320245377238666248026521863698157365356790349162864711782205289356s^{107}+216352869031916941103974168883097076410229240427531516289660856895884573411040138391989008s^{106}-4808372152188842416685049720889157609648589700259906966073914535288536332426660045281665576s^{105}+102730532296869024506560636386673680062422941215075875113492165335062102679114067711595589286s^{104}-2111527924549381695924838025670328572466346746690319171877130768902470515781848199414410720936s^{103}+41831905131678357252471218884771916619975870088217821433418140699364041647672903374550946065106s^{102}-799937696943205886362309315869391648647226626812787165601913537214769018438999347145254423256568s^{101}+14750154008419083794676619125813636103060922241286812273744920967608624399426277671233689970750816s^{100}-260975431069180213161379096976636911786047810516915330900553443161331183641061936532309426166815268s^{99}+4392344885718180821788009252695595386788391929454071852322904840485213772415318679286843092164107326s^{98}-69572449764366603777509810702402733716637668631656846420562636123468612787624389077681592721918843488s^{97}+1027097836001128208475480477047724529415634205788396193786255672967383685469857620307489972997358673526s^{96}-14074624684738338344020983555256300931561811169605846968675883195494170910423934868009815189998913131444s^{95}+180660626905403779319995260410889237792268816586814685932269189175254703565979443977412180365929218465280s^{94}-2245499788361448111606620260028829428953487326596025966685205057335044673291677551140402757171317681417428s^{93}+28671416193365649478884844283419050835810755858475230587266033381784144945601933009018387101662354025540734s^{92}-394217288266871322892333222337975011689661831536401018523536110370589489067761042286000949956400639666042316s^{91}+5703863195259166781025704560694159480317740632678599117279192239656528808349240434327194661001025914590969266s^{90}-80059468446622758198796944179244414275773216304886340967166998450686606958848716898739757721312778409184831024s^{89}+1004138763079804275697043916259570490495614067448884056635127265173895325706519305292407648765439933024547087131s^{88}-10573856961381771762913436897417012389280833263671551808955616987901353562852378162633710797010666619591470102324s^{87}+89592018192513343122608501728306468444080160150570013764942281507872879104525037337900571968575192851768883617072s^{86}-646389054343915846566881857708086454471929561189842086401340461316692430061948367583499858586482550023315255624968s^{85}+6456718039810642099097859415092985378580884294959550540027954767107746733301451637312622694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Found by Károly Hajba
in October 2024.
Improved by David Ellsworth
in November 2024.
Extends the $s(11)$ found by Walter Trump in 1979.
109
$s = 6 + {7\over 2}\sqrt 2 = \Nn{10.94974746830583}$
Continues a pattern found
by Frits Göbel in early 1979.
Explore group
122
$s = 8 + {5\over 2}\sqrt 2 = \Nn{11.53553390593273}$
Adds three "L"s to the $s(65)$ found
by Frits Göbel in early 1979.
123
$s = {}^{12}🔒 = \Nn{11.60139979378801}$
$196s^{12}-26488s^{11}+1651440s^{10}-62766716s^9+1618595997s^8-29815750560s^7+402059235258s^6-3996945599928s^5+29059125001479s^4-150620866376828s^3+528127756018414s^2-1124357849826920s+1098765914618225=0$
Found and improved by David Ellsworth in December 2024, by extending the $s(102)$ found by Károly Hajba in September 2024 and adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Improved by David W. Cantrell
in January 2025.
This new technique, which will need to be applied to about 20 additional packings previously thought to be finished, independently found by both David W. Cantrell and David Ellsworth in January 2025.
124
$s = 6 + 4 \sqrt 2 = \Nn{11.65685424949238}$
Continues a pattern found
by Frits Göbel in early 1979.
Explore group
125
$s = 11 + {1\over 2}\sqrt 2 = \Nn{11.70710678118654}$
Extends the $s(52)$ found by
Frits Göbel in early 1979.
Explore group
126
$s = {}^{9}🔒 = \Nn{11.77652079061690}$
$185761s^9-17452914s^8+733790386s^7-18120925928s^6+289457197920s^5-3096648733600s^4+22129577852576s^3-101471852915328s^2+269345322921472s-312506709170176=0$
Found by David Ellsworth in December 2024, based on the
$s(39)$ found by David W. Cantrell in August 2002.
Improved by David W. Cantrell in December 2024.
127
$s = {21\over 2} + {1\over 2}\sqrt 7 = \Nn{11.82287565553229}$
Extends the $s(86)$ found by
Erich Friedman in 1997.
128
$s = {}^{40}🔒 = \Nn{11.82509196821368}$
$6765201s^{40}-2845098648s^{39}+580218189444s^{38}-76444672534332s^{37}+7313700827489408s^{36}-541478198967270496s^{35}+32282461276159060998s^{34}-1592401650332728651212s^{33}+66264078654438595433370s^{32}-2360056578357132809629748s^{31}+72739322148379315505619294s^{30}-1956653332398832550681084364s^{29}+46240556849238042597672084213s^{28}-964953122823176982192672366676s^{27}+17849625763191276058228284471146s^{26}-293485855211047587757484370460572s^{25}+4296830596115061642805514450068373s^{24}-56062026207183424768909339152161084s^{23}+651753580603034236066787916795623964s^{22}-6743132871649823544913588155645442656s^{21}+61932541245281919280766873166108057833s^{20}-502864388097908644186877008296638914520s^{19}+3586164305875893817104519150255679222830s^{18}-22233865523258917656574596200774765950936s^{17}+117836973554812356574280571124747241305248s^{16}-517669550834165752088850944695261707233564s^{15}+1760771073593491738800908345083273891209716s^{14}-3688451224681060758343247748983014976447272s^{13}-3038946790141535208034747244682683990472658s^{12}+71735791138334020614589956733651574834185892s^{11}-381139020218879456095819418101710611273627456s^{10}+1267518021830549968006069595148713683301836788s^9-2664006348624383405491809555245054393205103691s^8+1793492920161456183420660943446317121911023248s^7+10946284003184957815907569175715318463243584102s^6-52105146308542520023979647560364421311730107464s^5+129236789770564934710391555764344704402747982753s^4-211311873022032856645395618371922980818558351272s^3+230091479010044538699295485926277667974383569488s^2-153189145317923709976188931598288062177638461184s+47601488782509082091988946077133401472942285824=0$
Found by David Ellsworth in November 2024, based on the $s(69)$ found by Maurizio Morandi in June 2010. Improved by David Ellsworth in December 2024, including using the technique from the $s(53)$ improved by David W. Cantrell in December 2024.
Improved by David W. Cantrell and David Ellsworth in January 2025.
129
$s = 10 + {4\over 3}\sqrt 2 = \Nn{11.88561808316412}$
Found and improved by David Ellsworth in December 2024, adapting/extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Similar to the $s(70)$ found by Erich Friedman in 1997. Quasi-extends the $s(19)$ found by Robert Wainwright in late 1979, resembling one of its alternative packings found by found by David W. Cantrell in 2002.
130
$s = {}^{8}🔒 = \Nn{11.91119052015898}$
$5617s^4-(259504+12612\sqrt{2})s^3+(4502202+422478\sqrt{2})s^2-(34773984+4731588\sqrt{2})s+100926915+17726970\sqrt{2}=0$
$5617s^8-519008s^7+20936788s^6-481754784s^5+6917560482s^4-63487737120s^3+363767813964s^2-1189915431840s+1701575795025=0$
Found by David Ellsworth
in November 2024.
Improved by David W. Cantrell
in November 2024.
Improved by David Ellsworth
in November 2024.
Extends the $s(88)$ found by Erich Friedman in 1997; adapts and extends the $s(37)$ improvement found by David W. Cantrell in September 2002.
Further improvement pending.
131
$s = \Nn{11.97350182495032}$
Found by David Ellsworth
in November 2024.
Fits an $s(n^2\!-\!n\!-\!1)$ pattern found
by Joe DeVincentis in April 2014.
Explore group
132
$s = \Nn{11.99790201730589}$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Bounds the $s(n^2-n)=n$ conjecture to $n<12$.
145
$s = 9 + {5\over 2}\sqrt 2 = \Nn{12.53553390593273}$
Adds four "L"s to the $s(65)$ found by Frits Göbel in early 1979.
146
$s = {}^{16}🔒 = \Nn{12.60090777851301}$
$4s^{16}-664s^{15}+51488s^{14}-2476628s^{13}+82758797s^{12}-2038683388s^{11}+38335080074s^{10}-561971068668s^9+6500206337793s^8-59622322943944s^7+433020664362230s^6-2468567342450368s^5+10847755732088938s^4-35571513994776316s^3+82161582314776868s^2-119456945705052640s+82323170428761425=0$
Found by David W. Cantrell in January 2025 by switching to the rotationally symmetric form of adding an "L" to the $s(123)$ found and improved by David Ellsworth in December 2024 and improved by David W. Cantrell in January 2025 with a technique independently found by both David W. Cantrell and David Ellsworth.
This is the first best known packing found that is doubly semi-primitive.
147, 148
$s = 7 + 4 \sqrt 2 = \Nn{12.65685424949238}$
Continues a pattern found
by Frits Göbel in early 1979.
Explore group
149
$s = 12 + {1\over 2}\sqrt 2 = \Nn{12.70710678118654}$
Extends the $s(52)$ found by
Frits Göbel in early 1979.
Explore group
150
$s = 5 + {11\over 2}\sqrt 2 = \Nn{12.77817459305202}$
Found by David Ellsworth
in January 2025.
Extends the $s(85)$ found by Erich Friedman in 1997 how the $s(66)$ found by Evert Stenlund in early 1980 extends $s(26)$.
151
$s = {23\over 2} + {1\over 2}\sqrt 7 = \Nn{12.82287565553229}$
Found by David Ellsworth
in November 2024.
Improved by David Cantrell
in December 2024.
152
$s = {}^{84}🔒 = \Nn{12.83100282216725}$
$15197358585941502961s^{84}-14964876021186235227298s^{83}+7272595222812913346894003s^{82}-2325304163405568837985711872s^{81}+550199014011902976417247729787s^{80}-102744907802613577303858434436162s^{79}+15770698108244892301462590278910373s^{78}-2046188803886728893706559592846166940s^{77}+229043365901827791271124033850409760904s^{76}-22465861798723536733581343377318969312880s^{75}+1954663063042408634836851737534542472988316s^{74}-152349951255709169941326823887066293346193624s^{73}+10723829667261655134952521210622665945643784968s^{72}-686326223380322052977518105478585500875790507640s^{71}+40166977336061074256998638756973363193038422762460s^{70}-2160190971013429916138221567315873809290109191227128s^{69}+107210371553542169414048201729669662441830314197024226s^{68}-4928367697648782759676964001463807238961374648608591036s^{67}+210520282427005491506665365123774698375707578804045991166s^{66}-8380048643861127985429658865920308979713621168800528636888s^{65}+311642979584387473990467919553614272825113220053872893167686s^{64}-10851807764621222577237269161423835499294065873227886719724876s^{63}+354529200442735318891489329639289130837080983716609686105844682s^{62}-10886472654344044554989820001276664684754352275834326626073676952s^{61}+314707830740258412075262705001018514633277114685570369659228230886s^{60}-8577131365576184973326383639361903919077060020501484349486798462044s^{59}+220676659864024529834222826550027169451751436159279143212643304100750s^{58}-5366108560854773099323062782542881689148622349128272868132841503304592s^{57}+123455035085747414638373053783111838415779807885794670228653620606756656s^{56}-2689771757042576617382266796760465801989789910549251473683839639432673824s^{55}+55545344660722669971771487790593823603421739027747117357084896573560148132s^{54}-1088014439077084998878807873686497414222608555337013172037085789785737027128s^{53}+20228760646280093419477169754640747236777672568049390223893488286064642987733s^{52}-357199319350132942687604624139422297924839641572382189922202437482576762356002s^{51}+5993578492259294457183648659781874954535555138522882079496926437004560861449475s^{50}-95607888374734158621452463758465962667527766889151119870233929745815458197870464s^{49}+1450449889734808340651325100183427866248693256913059099309145940155004921623663049s^{48}-20934174377970907243549394685205967930306737373954662348822439943609230423042271998s^{47}+287521214531466154844493934641558475944025274728304204144868014752283863593850841071s^{46}-3758693037139824522492196965950225707948971038134502497610180651441239588793706153212s^{45}+46776156166368620484370637225042926774946526631884418623684508412824264790410695322622s^{44}-554213498552523069017939780278197270272937025886205771218096298876624373232634030307268s^{43}+6251950732475223994427997225766260343105672057938693543298710223719438546214455983422174s^{42}-67148743958288503662262110524028512420572740818205612649788782970039163837877122850002224s^{41}+686624504976838308996418090639964921283797740974390943510927597462462930108635451388956672s^{40}-6683624210795197215545578694769827454378481281503786991205805051093122539577000607518492112s^{39}+61922038451063372274525115664711614781745351857035675107097735737909228871958371309771201252s^{38}-545914626236899516397451030889695660434986026741727925639735898323422010790358226899471142192s^{37}+4578596394614695256434177101284715486045596223316501591974779425322438106683698243945133312657s^{36}-36519557390596147521078972913970989194289986313602140505586138261168362886227188120613809866274s^{35}+276908452084060463131678792053749272309227620650243616383774652462199807972600272617027105313715s^{34}-1995125618573564239441569968811948768877908975264805329921557835273555692406902924736432832072656s^{33}+13652302794929101077954196396097847995447641729570461853834329101037063785834852682999232389861183s^{32}-88673260640330913380557229503418078388290914029958962882747927428883329136607168216172985057506786s^{31}+546323078740762777551499851151749299500660383154079131025685851053212263193896570529297338202937057s^{30}-3190532419019051137207920527310810985363082905053938554402580063971581164010258646801817355978728172s^{29}+17647597956569345600264156757559024534779396494222803173516820647844865659043181903919579547455993405s^{28}-92370185046480996167766003031798069104355227778256861958172638341670501748986559657158143287384201746s^{27}+457063596008246343843246947652497192740604188316726144667941077879214729203288916667939807194976754475s^{26}-2135754321030318339803946148640432723925062036840170746475770465651607920747496502359936750118636877936s^{25}+9413245107777963706229663058434971410824508757077884118452667020417078901491483099639931354293862639251s^{24}-39081567129013269416348118385885630160005422872090989882489179092615712699596245833330552131314742641914s^{23}+152623748753623567775238681507133061671421661548557325363509181608802271715863896420772452309156580126893s^{22}-559755021749304754461642532733408124950018980895075585999339578832360136778923426307553163340902938629628s^{21}+1924575434803718372008270042897715699164358183917612079945858652072146738653167981508516336683663859061599s^{20}-6191354447775618609836439234314634277309179603333612701552040527406315440826478331716065451128871996534670s^{19}+18595526459860121841165340574719138895821326772209658546058960991768583728860803392397281925779742313102481s^{18}-52017877357788563455349165010925854251068156661668439408177886419332674892024542973348383001969034647203800s^{17}+135157501868692938192366500327069236098923290104122566272833032709737094637647438296644161923551840062478047s^{16}-325194345405021237494009612258086437235941593129725154570748027251348017358438973985857832818544091573466890s^{15}+722027337085845458293091421627894365163696335388182663348454109292879460761558869358682885259404296739097605s^{14}-1473490515445565711145580914132319537412939207599828450536062468067653025235732598973121176466499270242277300s^{13}+2751263110572258451275788319799047606274296134838009812884081346353904687815554600474281715625532788431857500s^{12}-4675004465981033004681427309219791294001064124937869088877001968417898773888626422941020050924220157893424000s^{11}+7183663179262068580520363661753568724897226652642552988417888182903983412500744464905921688273238836620034000s^{10}-9906503785168325598131719638667321753622423751680769939351178446313898592975073862661742325684296198395800000s^9+12146944042621050502155105222176120253288451344101362562568904240946898405180038535485859612413971290688600000s^8-13090089533045063828551500666675702412696959576885680069538751766268511536417556308969034651288724717832000000s^7+12214708422409722054795434067910885386456675508392415014637505990855532116097110284775444747515260212130400000s^6-9676741582044228838733364917804914719594151753924645775721115515700953860674142281619525345349780994608000000s^5+6333689917226114819893572648325901156172023728019394254226767788794528703035140896587138185275926267216000000s^4-3291213620680948097795240144789699630837845626207176614176801595943955328302192020992095429065402069760000000s^3+1274139733072931259621243156734865080962818490295409701367431118912261529814385684303575896038896856000000000s^2-326961724135799194553082826640306328403985650565477078726674395723377915426470127881664726913341548800000000s+41749176420191321903553288073015514269407918838782863228971623490248342604054718097305744082801772800000000=0$
Found by David Ellsworth and David W. Cantrell in January 2025, based on the $s(53)$ found by David W. Cantrell in September 2002 and the $s(69)$ found by Maurizio Morandi in June 2010, adapting and extending the $s(69)$ improvement found by David W. Cantrell in August 2023.
153
$s = {}^{4}🔒 = \Nn{12.88166675700900}$
$23s^4-1110s^3+19960s^2-158164s+464677=0$
Found by David Ellsworth
in November 2024, based on the $s(70)$
found by Joe DeVincentis in April 2014.
154
$s = {}^{4}🔒 = \Nn{12.93786550630255}$
$s^4-40s^3+599s^2-3950s+9446=0$
Found by David Ellsworth in December 2024, by combining two slightly modified copies of the $s(41)$ found by Joe DeVincentis in April 2014 that fits an $s(n^2\!-\!n\!-\!1)$ pattern.
155
$s = {}^{4}🔒 = \Nn{12.97970624703929}$
$s^4-28s^3+299s^2-1376s+332=0$
Continues the $s(n^2\!-\!n\!-\!1)$ pattern found by Joe DeVincentis in April 2014.
Explore group
156
$s = \Nn{12.99016864798692}$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n)<n$ for $n=13$.
Improved by David Ellsworth in December 2024.
170
$s = 10 + {5\over 2}\sqrt 2 = \Nn{13.53553390593273}$
Extends the $s(65)$ found by Frits Göbel in early 1979. This alternative, converting the $s(65)$ augmented by five "L"s into a primitive packing, found by Károly Hajba in November 2024.
171
$s = {}^{8}🔒 = \Nn{13.59569998314336}$
$96s^4-(5968+928\sqrt{2})s^3+(133541+34462\sqrt{2})s^2-(1291883+427186\sqrt{2})s+4598280+1769696\sqrt{2}=0$
$9216s^8-1145856s^7+59534528s^6-1714063968s^5+30175010609s^4-334467918206s^3+2288154693249s^2-8856922100656s+14880531093568=0$
Found and improved by David Ellsworth in November and December 2024, by extending the $s(102)$ found by Károly Hajba in September 2024, and adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Further improvement pending.
172
$s = {}^{8}🔒 = \Nn{13.61898898660160}$
$54s^4-(3936+864\sqrt{2})s^3+(98658+33466\sqrt{2})s^2-(1047336+434398\sqrt{2})s+4049739+1891234\sqrt{2}=0$
$2916s^8-425088s^7+24654168s^6-774089568s^5+14674175968s^4-173849337008s^3+1265419604436s^2-5196681822080s+9246853882609=0$
Found by Károly Hajba in November 2024, extending the $s(102)$ he found in September 2024. Improved by David Ellsworth in November 2024 by adapting the $s(37)$ improvement found by David W. Cantrell in September 2002. Improved by David Ellsworth in December 2024.
Further improvement pending.
173
$s = 8 + 4 \sqrt 2 = \Nn{13.65685424949238}$
Adds an "L" to the $s(148)$ that continues a pattern found by Frits Göbel in early 1979.
174
$s = 13 + {1\over 2}\sqrt 2 = \Nn{13.70710678118654}$
Extends the $s(52)$ found by
Frits Göbel in early 1979.
Explore group
175
$s = 6 + {11\over 2}\sqrt 2 = \Nn{13.77817459305202}$
Found by David Ellsworth
in December 2024.
Based on the $s(233)$ that continues a pattern found by Frits Göbel in early 1979.
176
$s = {25\over 2} + {1\over 2}\sqrt 7 = \Nn{13.82287565553229}$
Extends the $s(86)$ found by
Erich Friedman in 1997.
177
$s = {}^{32}🔒 = \Nn{13.82302875075647}$
$2401s^{32}-931588s^{31}+169658874s^{30}-19231837912s^{29}+1515206475113s^{28}-87554613482844s^{27}+3800602014647796s^{26}-123749571598485028s^{25}+2895774212866682688s^{24}-40503170127651197920s^{23}-85797782465034115616s^{22}+22358489056150565928884s^{21}-670371983793922205889766s^{20}+11585516453329663611601440s^{19}-107946265554474207035275274s^{18}-325416702082583543878844088s^{17}+31678001381593454789856242308s^{16}-627531463707625262161828471384s^{15}+7738051841906036676459384893372s^{14}-65825971528188631650991553430380s^{13}+390859965437244296776723867974104s^{12}-1981263185614787515387345660456708s^{11}+20697660792797450992561898119685608s^{10}-341458570720939415832614447938585072s^9+4331777593881436240458282284914909233s^8-39425956807257141453494313251818730308s^7+265332233224572263186725591178500406376s^6-1341713349412185465951061216890867086592s^5+5078477010422928472451310243101300746580s^4-14041240100101609173845558593200611383568s^3+26892648324465918884312724426364749628480s^2-31980711702815550619890642368200074811200s+17821422876028786503270705680802036472000=0$
Found and improved by David Ellsworth in November 2024 and December 2024, based on the $s(53)$ found/improved by David W. Cantrell in September 2002 and December 2024, respectively.
178
$\begin{aligned}s &= 13-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{13.84666719284348}\end{aligned}$
Extends the $s(54)$ found by Joe DeVincentis in April 2014.
179
$s = {25\over 2} + \sqrt 2 = \Nn{13.91421356237309}$
Found by David Ellsworth in January 2025, using a computer program he wrote.
Improvement pending.
180
$s = \Nn{13.95491767825175}$
Found and improved by David Ellsworth in December 2024, by removing 2 squares from the $s(182)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Possible improvement pending.
181
$s = \Nn{13.97854770217285}$
Found by David Ellsworth in December 2024, by removing 1 square from the $s(182)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
182
$s = {}^{66}🔒 = \Nn{13.98264294789927}$
$1713850177070388831714241s^{66}-923663845843030533029467964s^{65}+244356526675459843171256658812s^{64}-42301149163464927098830722044712s^{63}+5389637680061232065659444372903884s^{62}-538990344040466547260756897354234800s^{61}+44060430495129161810848298418942728768s^{60}-3027597018002804743276382310802134926912s^{59}+178476725866222249279843650493042891414174s^{58}-9167125837455703044437102107796599060405072s^{57}+415277865358575078055753826628996914179647584s^{56}-16755219749633813087434458219665047861571005696s^{55}+606951588795740859789587928711977301863618562444s^{54}-19872617896564458386479749674438181110594571783440s^{53}+591430450596695946776559043737897061925817091883896s^{52}-16076403382596088035670122941655178909884706102259360s^{51}+400780790911848317908826372889780636781368953842134545s^{50}-9196292015835695622526434508695387052579786474968530764s^{49}+194831950657120243957930949495672896263326187482945017636s^{48}-3821488517527886494201924904813013938910357960132992875112s^{47}+69561259675051249201492552497458936188600783387550785194512s^{46}-1177541097875302951212548144402084258394462103241418878091904s^{45}+18572022118855878686507143350115259541331165972203297602035016s^{44}-273350438905317141283665359004770430754125267452024933490938688s^{43}+3759875778293105916627657039864999051693799407851712882736113208s^{42}-48390183666058035250293916819527929345794012592770993441540548200s^{41}+583359495992769368180893839833879135938973736557186058881098499424s^{40}-6593404023892398478887639745053689330072639258527396503597740822704s^{39}+69922254584057803430223785412744087886168418412416437299744434763920s^{38}-696199801753965501285725548974198631608011266843672974698091490432160s^{37}+6511650429478482167950610493981767940220071476762446758258297829362464s^{36}-57234890555844218227733552885344124231228022552073037991205841355754784s^{35}+472897245542288005015724033828084119792469894384102779109267997968480096s^{34}-3673527397539404818977007733654565672521620497606425069188768378430791392s^{33}+26830967247160316756709981320435477112019319138841629014760220636370474816s^{32}-184248363314616695085417127987503784871627748574838179238693969981716890752s^{31}+1189362835357759435647680170936163404089852639152762150758843550851500029312s^{30}-7215198553515381078457422107373261572387237955159216402793836179726031898368s^{29}+41118237990763833027717928080095280450530652728789049212131791394290259869888s^{28}-220013586830299858346544628394840804319064288631784649846740295442808772668160s^{27}+1104624680933328901706883047576578922329679532689002041544873488282388304886016s^{26}-5199881073657329493566722212089680068736612815975219398932756978414737762064896s^{25}+22929072742861089738590065098448598702132646005025936214483918913337650353069824s^{24}-94608008640637296299957783267672779275948371642923398380136113133454728406140416s^{23}+364819925689129881972579265818449668453976797240020471862122424163060417660220928s^{22}-1312862171184299432821030617749268935136513935049032859605965972625387377719365120s^{21}+4401912097721267325119409154299200433015462297229066215750087595154529481296731136s^{20}-13725788367315319506315635643739707608001009560356143859397586956998117539703622656s^{19}+39717900153429166873869781810993511500356117237564376538643805576134915824234523648s^{18}-106399332388497647615120413383665293526457978124604308307567938191333861608693833728s^{17}+263146214957976354333140167648787212861652057702279153736060651315259635818278014976s^{16}-598948423050050797962529487281503994476619892474808687630034845197011146318218905600s^{15}+1250084500161595434165407078512122830805401588230922174957693497088606844442474217472s^{14}-2382458928043568775151925441566626859323222839887782280481843477807775050701314756608s^{13}+4125994826986881535378038074633666561772029457781662960093867812493907378777176526848s^{12}-6456003870217606716135295888226937623998759028219550792812724057683607497186984583168s^{11}+9065372956178732821827510381589091533382082905889742319984764707994104046708843806720s^{10}-11330803663084115039545790076138860721501300902182053330467921505356766127529332162560s^9+12481974955426380241965795213639280113780047248776326121594789973305180053750293086208s^8-11970325319419753577043006848391910395086707654239813304187044220003154526297518047232s^7+9838402392037009878211651167751684462461327699694726737875855220251210658084493393920s^6-6789017987208597545069669454098514525060401317357596876693764512756899474772483768320s^5+3824088158296275561318991814461422505555214264984822149493993857503600579756629164032s^4-1687927982575032415641479063772412094421237083850884119669396632875631662093020692480s^3+547314545007055176499085872328704154179592545295820829835019843713968328240803086336s^2-115875113076825478429958642319077276340668682757823451130118465600944508946094227456s+12012260314488957307342435041239878634388761717780836316682092965799623955237044224=0$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n)<n$ for $n=14$.
Improved by David Ellsworth in December 2024.
197
$s = 11 + {5\over 2}\sqrt 2 = \Nn{14.53553390593273}$
Adds six "L"s to the $s(65)$ found by Frits Göbel in early 1979.
198
$s = {}^{8}🔒 = \Nn{14.59153604974246}$
$7889s^4-(496196+66432\sqrt{2})s^3+(11353289+2588872\sqrt{2})s^2-(112836070+33552520\sqrt{2})s+413156140+144732976\sqrt{2}=0$
$7889s^8-992392s^7+52797090s^6-1566650388s^5+28530003977s^4-327826283468s^3+2327673598588s^2-9356471793680s+16326876326032=0$
Found by David Ellsworth in December 2024, by extending the $s(102)$ found by Károly Hajba in September 2024 and adapting and extending the $s(37)$ improvement found by
David W. Cantrell in September 2002.
Further improvement pending.
199
$s = {}^{8}🔒 = \Nn{14.61898898660160}$
$54s^4-(4152+864\sqrt{2})s^3+(110790+36058\sqrt{2})s^2-(1256676+503922\sqrt{2})s+5199723+2359962\sqrt{2}=0$
$2916s^8-448416s^7+27711432s^6-931104720s^5+18929518528s^4-240795061296s^3+1883132387964s^2-8311787117640s+15898277993841=0$
Adds an "L" to the $s(172)$ found by Károly Hajba in November 2024, improved by David Ellsworth in November 2024 by adapting the $s(37)$ improvement found by David W. Cantrell in September 2002, and improved by David Ellsworth in December 2024.
Further improvement pending.
200
$s = 9 + 4 \sqrt 2 = \Nn{14.65685424949238}$
Adds two "L"s to the $s(148)$ that continues a pattern found by Frits Göbel in early 1979.
201
$s = 14 + {1\over 2}\sqrt 2 = \Nn{14.70710678118654}$
Extends the $s(52)$ found by
Frits Göbel in early 1979.
Explore group
202
$s = 2 + 9 \sqrt 2 = \Nn{14.72792206135785}$
Extends the $s(18)$ found by
Frits Göbel in early 1979.
Explore group
203
$s = 7 + {11\over 2}\sqrt 2 = \Nn{14.77817459305202}$
Found by David Ellsworth
in December 2024.
Based on the $s(233)$ that continues a pattern found by Frits Göbel in early 1979.
204
$s = {27\over 2} + {1\over 2}\sqrt 7 = \Nn{14.82287565553229}$
Extends the $s(86)$ found by
Erich Friedman in 1997.
205
$s = {}^{40}🔒 = \Nn{14.82445114612408}$
$6765201s^{40}-3656922768s^{39}+958435862652s^{38}-162258762163956s^{37}+19944495118895680s^{36}-1896817614997924732s^{35}+145244607337067116310s^{34}-9200231755130148537584s^{33}+491532552940826676421598s^{32}-22471266124455622725351620s^{31}+888775367178021320983990528s^{30}-30670374642432504931208393536s^{29}+929494097506723220926745638527s^{28}-24862568075564651560878275579812s^{27}+589158528037990588430776578667278s^{26}-12400310845434669601706739856078348s^{25}+232178750736807601084495428509773447s^{24}-3869279274647141302600286447060842028s^{23}+57359828211423634121225283370282978472s^{22}-755020486735335189866154838905984601284s^{21}+8794114159490040759213225924279409129174s^{20}-90126731671943479290165241436087360926864s^{19}+805381183069421598181526905692816890290502s^{18}-6181562346162193722542638238059633411805108s^{17}+39656170956144484959733239448754004343189341s^{16}-200557184022962853819962178120475887680432104s^{15}+668299828757988666056469128930860611880478162s^{14}+47787548247188382784775375655144589392168024s^{13}-20274687460176342070123115001595419340631113986s^{12}+171806640590364466782586102889249290168160938968s^{11}-887645374096288133345472520371583078066772417438s^{10}+2995081869452847889056141420143309758645762048480s^9-4490901292932914262604076259256067123710050815195s^8-18138284909745955939878418372807115317752603411176s^7+165869320264318841451647718375161237581492991097810s^6-695527958949195756061996697585676748194320009594040s^5+1948735509481744597514888658808885791132097191929625s^4-3838064311683628303762259793886602147590339042385000s^3+5169742707762800848775847149992971475362078199650000s^2-4315559969700226767159393981391684637349630444000000s+1694040395896101772224866098180283214283057728000000=0$
Found by David Ellsworth in December 2024, by extending the $s(128)$ he found/improved in November/December 2024, based on the $s(69)$ found by Maurizio Morandi in June 2010, and using the technique from the $s(53)$ improved by David W. Cantrell in December 2024.
Improved by David Ellsworth in January 2025, using the technique from the $s(128)$ improved by David W. Cantrell and David Ellsworth in January 2025.
206
$s = 13 + {4\over 3}\sqrt 2 = \Nn{14.88561808316412}$
Found and improved by David Ellsworth in December 2024, adapting/extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Similar to the $s(70)$ found by Erich Friedman in 1997. Quasi-extends the $s(19)$ found by Robert Wainwright in late 1979, resembling one of its alternative packings found by found by David W. Cantrell in 2002.
207
$s = {}^{12}🔒 = \Nn{14.89564154083791}$
$16s^6-(1144+96\sqrt{2})s^5+(33985+6024\sqrt{2})s^4-(537992+150024\sqrt{2})s^3+(4801816+1850340\sqrt{2})s^2-(23022312+11275344\sqrt{2})s+27067200\sqrt{2}+46636900=0$
$256s^{12}-36608s^{11}+2377824s^{10}-92660208s^9+2409377665s^8-43965076368s^7+576052330352s^6-5445742834320s^5+36732970030440s^4-171582571927744s^3+523309619017472s^2-926610560558400s+709733809930000=0$
Found by David Ellsworth
in November 2024, by extending the $s(88)$ found by Erich Friedman in 1997, and adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002 and the $s(130)$ improvement found by David W. Cantrell in November 2024.
Further improvement pending.
208
$s = 10 + {7\over 2}\sqrt 2 = \Nn{14.94974746830583}$
Originally found by David Ellsworth in December 2024.
Alternative found by David Ellsworth in January 2025, using a computer program he wrote.
Extends the $s(89)$ found by Evert Stenlund in early 1980, which continues a pattern found by Frits Göbel in early 1979.
209
$s = \Nn{14.96442179744201}$
Found by David Ellsworth in December 2024, by removing 1 square from the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019 and improved by David Ellsworth in December 2024.
210
$s = {}^{66}🔒 = \Nn{14.98264294789927}$
$1713850177070388831714241s^{66}-1036777957529676195922607870s^{65}+308070885285072811862199123417s^{64}-59939613454150601563431685241960s^{63}+8588813678023291261040553534326772s^{62}-966583568773741127569549391339079520s^{61}+88973510619748651618414859862026730876s^{60}-6888549315978477501144600760825939298456s^{59}+457813058489637019372876565898875762561962s^{58}-26525924226609242481987455544648574755740060s^{57}+1356297999041102109336083334683724973134359610s^{56}-61799914557147552310687442895919575623521766840s^{55}+2529586572366492011492859050137381026019860276228s^{54}-93635049365714878103413599709118844924967191715568s^{53}+3152090707330317380212505369273541211594578660668828s^{52}-96964556710709267169581794961613905049794026876686584s^{51}+2736977327538183203599671149196691182209406477480426999s^{50}-71141341918871267999443622612794042553629895963414923554s^{49}+1708099525395866416691912276683615496482061285927149098767s^{48}-37985865900286624496819504298337826100292397570681660970624s^{47}+784296971154264745366602352186813069922416329570936529960120s^{46}-15065885306353527422811618762658136922751746094197077043607808s^{45}+269749236797310845207931452385959355761508726043413698724913752s^{44}-4508953939245292278547355693056226796525821395294109220913219760s^{43}+70461706096440343504339359622207886316103484007784533453402451564s^{42}-1030684846592380569306994628265946835298274035988655898515756518536s^{41}+14127213285791424916401341828825119374306871082227591450747564708428s^{40}-181611012865775124333222503250168271689890578495810300493810490865200s^{39}+2191389437227659168703888781115852382017210534407970049772895556135144s^{38}-24835252575107480313771638509524748878675003716854534342036362523912096s^{37}+264493817830994381831108752939852505433266002142853663596745464038105112s^{36}-2648106506791820424837514470078773670571332661118776845107705934142835760s^{35}+24931817840154101718618675302330328982025705361265314835179979339556004095s^{34}-220774232592783180535637731350600608019775280385941635854721272379834520226s^{33}+1838859783489983364796248010427404448952293664994580697429468489690817448391s^{32}-14405793276814386692861915917557813798574320839280787544681770605702716243496s^{31}+106132652991766969969790599313795738256162256172065649640195937493783785627012s^{30}-735141103620029965659773734189613183265405874983552791121595763935147775210784s^{29}+4785629939469775130214760518204470967279654407141021153131677479220692014706252s^{28}-29264396180717787590970697217467423077531807932966908615153820237900145532452792s^{27}+167997715129752800946484031210802390736511802747887658272905155103288606648529418s^{26}-904700154474475752208581796921737926689902979381082659180178913034340167719783580s^{25}+4566216117618812719619282953953565095633492925168145038704985120782850946704225818s^{24}-21577735591142156619368360651289725338415069076557628274035844600998082681026841624s^{23}+95351638011084223101955380792874473279508189366232625045913569058491535016623736724s^{22}-393477264331394419045246438963291356261882874125326625922439475690536334420518557680s^{21}+1513871326020646900995786895290089433666501518786395002635104303984667876045713548492s^{20}-5420589319604280486120080489351603688574661016232355428256686514216840385980014845144s^{19}+18025671901154216272856257915455921861722569907273247804240829505876591992203422527785s^{18}-55538726517617123321306980401388416698101944528011997777048904298548850658621880672382s^{17}+158119388173030536226827428726717197786937874399240446800307626931688853687437117891505s^{16}-414679766978138921331059461345942561913790083283653894727325899938505427858892965125872s^{15}+998228943688800328823203749490058494562116495933106174279441691815211346156075844428112s^{14}-2196572339864925849617348696491086772493215350378298856026223386996595289374884651752448s^{13}+4397165113585946993607002307549324635278671461053934532945038939803841295777334091620192s^{12}-7962739349659864081806356456004225116652369166110620089333680683881143975151724678690048s^{11}+12957137888727586584941034560243767737163859971700725359751881246527790566047851156973568s^{10}-18794133216992559109409136095289044747144304027555381568607235733444082266478813293896192s^9+24062707252931451750084904210379578567640678822720136431418887027294986269907390293532928s^8-26864643906507885513435835499475292614570149380412717542767363827389611363117057750564864s^7+25750555425531123093287716207567520607853292058501569356839679301847154190381313748566016s^6-20763167580966447999929009292788490161886160932306769296176687436564687869505524614365184s^5+13694576920953853335160219343243224031929703522356095792630808590038480485959704606801920s^4-7094101031359142126885338736264230663456345418028425142721873216376995826523389107896320s^3+2706320051143905447935556673344831161461373449684695751208915990592616770507314550538240s^2-675931907390084983300050927121665770468081954992407462935298185140040600624082498617344s+82905798822484597710700208785408047898314652058813563312936555293758738816713843802112=0$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n)<n$ for $n=15$.
Improved by David Ellsworth in December 2024.
226
$s = 12 + {5\over 2}\sqrt 2 = \Nn{15.53553390593273}$
Adds seven "L"s to the $s(65)$ found by Frits Göbel in early 1979.
227
$s = {17\over 2} + 5 \sqrt 2 = \Nn{15.57106781186547}$
Found by David Ellsworth in January 2025, using a computer program he wrote.
Continues the $s(26)$, $s(85)$ series found by Erich Friedman in 1997.
228
$s = {}^{12}🔒 = \Nn{15.60902282132495}$
$24s^6-(3008+632\sqrt{2})s^5+(143990+46836\sqrt{2})s^4-(3498934+1393560\sqrt{2})s^3+(46342192+20798868\sqrt{2})s^2-(320302448+155625644\sqrt{2})s+907787225+466763736\sqrt{2}=0$
$576s^{12}-144384s^{11}+15160736s^{10}-915791264s^9+36096990788s^8-988137514952s^7+19384693200660s^6-275725098032832s^5+2830331680270636s^4-20490408263563724s^3+99460046477154240s^2-290971312817869664s+388340875383845233=0$
Found by David Ellsworth in November 2024, by extending the $s(102)$ found by Károly Hajba in September 2024 and improved by David W. Cantrell and David Ellsworth in November 2024.
Improved by David Ellsworth in December 2024.
Further improvement pending.
229
$s = 10 + 4 \sqrt 2 = \Nn{15.65685424949238}$
Adds three "L"s to the $s(148)$ that continues a pattern found by Frits Göbel in early 1979.
230
$s = 15 + {28\over 41} = \Nn{15.68292682926829}$
Found and improved by David Ellsworth in January 2025. Uses a rotational symmetry technique found by David W. Cantrell in January 2025, and the technique from the $s(293)$ improved by David W. Cantrell in January 2025.
This is the second record-setting packing found with a rational side length, thanks to the Pythagorean triple $\{20, 21, 29\}$ determining its tilt angle.
231
$s = 15 + {1\over 2}\sqrt 2 = \Nn{15.70710678118654}$
Extends the $s(52)$ found by
Frits Göbel in early 1979.
Explore group
232, 233
$s = 8 + {11\over 2}\sqrt 2 = \Nn{15.77817459305202}$
Continues a pattern found by
Frits Göbel in early 1979.
Explore group
234
$s = {29\over 2} + {1\over 2}\sqrt 7 = \Nn{15.82287565553229}$
Found by David Ellsworth in December 2024, by extending the $s(128)$ and $s(205)$ he found/improved in November/December 2024, based on the $s(69)$ found by Maurizio Morandi in June 2010, and using the technique from the $s(53)$ improved by David W. Cantrell in December 2024.
Improved by David Ellsworth in January 2025, adapting the technique from the $s(128)$ improved by David W. Cantrell and David Ellsworth in January 2025.
235
$s = {}^{83}🔒 = \Nn{15.82660563342856}$
$15197358585941502961s^{83}-19054029395700781380870s^{82}+11796924216629042649354579s^{81}-4808108614769876120043468180s^{80}+1451030856404178122570781430789s^{79}-345799767275884788306353984453266s^{78}+67774498813564804634645374167057335s^{77}-11234508000945633407554425804213908384s^{76}+1607532344825153174801872971520978793189s^{75}-201668659118682967539387133057113450996510s^{74}+22454296891522560058771023859593458711158383s^{73}-2240889622579706580689718577985790904704763388s^{72}+202076916434677863681321953179975198173682882349s^{71}-16577690712578589166013893731214562354253387341274s^{70}+1244306501679531491606992301185434118328699167129267s^{69}-85872410283218882421430889383517071885115187773104056s^{68}+5471931933759042230241340087362460711039430660491160647s^{67}-323138905823957289824620179077427648515971541643473332874s^{66}+17742026808852576915081042170482522649963659530339729297241s^{65}-908282679855850349803743965878897364257354889699334907097884s^{64}+43465080817124537825282652035527867092782838207883438915180955s^{63}-1948672651714009470065517948906139063310384408556369781794367782s^{62}+82014228252452305308486172380001566000442685829960411231106701149s^{61}-3246186057482410594749950508613065532781882056581161088848208781448s^{60}+121030129123359970782879324721925653713348068894577966308153780376769s^{59}-4256774383890838346683359771242014391309685520375828879287765027421558s^{58}+141417395891604974478673542444206052527574423688874097457316926184111667s^{57}-4442950325166710767652318494176477005346284036871039689545149858425019188s^{56}+132143761554097136892656409949813501644806291750873673851743056646810847307s^{55}-3724278121653142074595785794031359776637918996855481246500264411247627583870s^{54}+99547162759226989443922275915301342990394148481013546431915854939056693130149s^{53}-2525451166316636919882169794346033300450573261760214407544561168810183261623712s^{52}+60851054822925884393362872088231784921476592249704525359339306693211978978637574s^{51}-1393407983201463082750103812667205031231534119903960161407503385026149533812640388s^{50}+30338905815230015350337507443629547938133322986372599251613731735431147148807587298s^{49}-628396204847897965060061083685759495211044889335692127791727849670706521484518435856s^{48}+12386583193977551660461241876390719826604576164873235895959785437607584760441587730192s^{47}-232433724444205471983096462597776075369365835330568989261391110164425311728212391035632s^{46}+4153323734217430140145070137613411920743170003732119525339537946385437046481746033523872s^{45}-70686380467717783902448903568646078963423628086622300763415121270383317250590261969131336s^{44}+1146011275499749445486974909121594149473099806147735475740109750306258973873693081841338630s^{43}-17701136698331714953844021503637463123650314086731790857767064280112784186506532020777615276s^{42}+260492277459372683985340515012561460001230750454462824800331349575718851511822995239394574714s^{41}-3652320068116991521691903234094675077664783199210342925565230028260558957233172085378428693648s^{40}+48786365568085939831648565613081789083584480449521988080326278540514634832953535303132015187720s^{39}-620776840486987337546656613456515581406101803594950169186131730819070047268868384631491241542280s^{38}+7523288801124157164696453440451668126537802179655932385275366967957982556047497857359919983352588s^{37}-86819738273128010242526385979967991470567862056150689620462183702554367768947413111616038233016416s^{36}+953774954925025648253304474733609592484747994431472390301868754639755982678716269084529930641443023s^{35}-9971081293651892380457737953978637874233272651346699205006125992828217866601482502757629011345866570s^{34}+99159113935527095843791193320314995744620431737333925602216138254954437828859473360567133620147394445s^{33}-937591882456243116942831217151994956616595933093755511723294563972551195048414611565248985607152821236s^{32}+8424667140126948167633953265193969962679276499025327494839034104790577492192909219817624591386758730279s^{31}-71892659494830178227616871456048478350079377496302951700870381917053589308972575388983039596535479420278s^{30}+582251421494115025057046976615977029976720394400712813199098003690123303907349827395875756580289035731733s^{29}-4471937667537034121585882112126200967254037591271575148365797097846048547850509627026833861786334101222136s^{28}+32543614650180016207874244187164028503629361855235911764700956406683943044733068359145297584031071397155994s^{27}-224183523933854882502864682078640188285826762871716486751957673723635189271192115484721934301078509843540820s^{26}+1460312333249097739412739848153412311136035088888884922686424538738960526601793346522957489286185673762602902s^{25}-8984148217876593612167877570397820635649870435669726928588777366245474742877218816201439836639845502892953560s^{24}+52134501077125370653590265849501908514957010403776225078820415168994018070692909692704843148671899014807836538s^{23}-284941297083362220078712187212822093342654560298933781771361807935255252780142074341846765908056419933065354916s^{22}+1464404571156043076914637252120097347255977567642248877431122565189765063022644175773142656036823662408630919410s^{21}-7064080779433205483997411627458329256081928935674885720573960927817446941144054662390438254917264293853571028424s^{20}+31919881457624033726193536951838036401111886302890312451699329606929040067283389942070240214272698940267541956493s^{19}-134801722329859078816742606094646334228067206837309021726684217834848607820988762454709381034223562103046457675998s^{18}+530707012396024913429576740380121941086737225273466472672403349930854389215481675641529836588022052334151834279003s^{17}-1942205476550803484882315247161577147422888428395169781637047795709098197490266699760943181834041758117906367397268s^{16}+6585767384476720675187977121595259536735035857996075281802198937005198489937485851461054520341053364378881846325263s^{15}-20614892940117095331097736607876870976314189060882921080291907574510740821764737500291921914248821586266401315666942s^{14}+59316353011229384909666880954476222015479423693489595098884075044567497728679992849404330173399583652230381335502305s^{13}-156117670782437502269990886042735284992957692699464203987769629925609348266900609376317045942894365095896351357414520s^{12}+373698270769972479755939957244422481116455737532015818487606025512111631824244935754777237146223432771503247544608560s^{11}-808048121315624822538485230387020159879703732818682364581834764813478780649668941927219429385916131623619099840318128s^{10}+1565583417239292630942979288694634658573835168674069318551077876822765011658251873382962913367365429573513955164437064s^9-2691230551310123431856658453025381333299196967548104418289076204697729928184484894708129837595375747556018998818512192s^8+4054542066273187475893579325962915065845321084583224373643180486702374676704731329107411831884877578115801740253541904s^7-5270878662724145249388262906951781140123258747428763886318687210813398134336463475140327838867254422860035408448158432s^6+5792836201656880167367652921094881746142782983189532980946755918671194010952081295188183352763611672876752443571623568s^5-5233614558124453824846563012926178182173958634839557295689659286203290796724301748107667652717293231008062406343836800s^4+3732102862328364130585244845143109502510706215299361787475517662634187688740089578345011212002758654232473965037848832s^3-1969628432244786328367806409266293170003496844663625159204453312874075246094750590119459796978694116823499446641506304s^2+683920819913111461189983402245349089127449075173547013376466597761071475636877690918573610486424418270329963001143296s-117203977757280124647356183279343983773836101226288348510440504667964165618210763520372674407323269025478320565518336=0$
Found by David Ellsworth and David W. Cantrell in January 2025, based on the $s(53)$ found by David W. Cantrell in September 2002 and the $s(69)$ found by Maurizio Morandi in June 2010, adapting and extending the $s(69)$ improvement found by David W. Cantrell in August 2023.
236
$s = {}^{12}🔒 = \Nn{15.87607676541001}$
$6596s^6-(619856+19000\sqrt{2})s^5+(24226944+1444540\sqrt{2})s^4-(504036792+43818500\sqrt{2})s^3+(5886496273+662906450\sqrt{2})s^2-(36585732598+5001593450\sqrt{2})s+94529982167+15055629320\sqrt{2}=0$
$26384s^{12}-4958848s^{11}+426380416s^{10}-22179443008s^9+777414394344s^8-19344554351312s^7+350411708641272s^6-4655972006018640s^5+45039272938487593s^4-309348405383979564s^3+1432057511182322314s^2-4011939813305628468s+5144071303851334561=0$
Found by David Ellsworth
in November 2024, by extending the $s(88)$ found by Erich Friedman in 1997 and adapting the $s(102)$ improvement found by David W. Cantrell in November 2024.
Improved by David Ellsworth
in December 2024.
Further improvement pending.
237
$s = {29\over 2} + \sqrt 2 = \Nn{15.91421356237309}$
Found by David Ellsworth in December 2024.
Similar to the $s(70)$ found by Erich Friedman in 1997.
238
$s = 11 + {7\over 2} \sqrt 2 = \Nn{15.94974746830583}$
Extends the $s(109)$ which continues a pattern found by Frits Göbel in early 1979.
239, 240
$s = {}^{66}🔒 = \Nn{15.98264294789927}$
$1713850177070388831714241s^{66}-1149892069216321858815747776s^{65}+379137661154317748641195681912s^{64}-81891074048159729003070697178408s^{63}+13032601897654000375127635736914044s^{62}-1629704803955659741528480932743629600s^{61}+166761277204901256215813369062945614448s^{60}-14358776608758577299719024788683561291200s^{59}+1061744286766486909447032265740305813536222s^{58}-68474327034163224324683196882521071680487400s^{57}+3898680014040310378546424786982864974691587032s^{56}-197893479763233829851953449206470703678382928256s^{55}+9027058972990094483551124675146985797402686345676s^{54}-372525707164344655382696831764231550931861454524704s^{53}+13986254561093171332246318032879802874415524621400936s^{52}-480021129624779505518793563427218381659637764171707744s^{51}+15122329258714384953594013673578917012766694983733804881s^{50}-438855029419115553162393305979405927798765869173444577904s^{49}+11768226418611255713545406862693986409497534694395932490528s^{48}-292389611865138370972616101847672913969792672213028811518184s^{47}+6746862076174715126011529791080074820752050442542414504370560s^{46}-144888150698948224043410517981223612722489152566776581298794208s^{45}+2900996894928148652325377674448953709338108189036698898804529608s^{44}-54242556288254952045704327412146319996536264265171980175919237760s^{43}+948462355395453479168602938292256298703061926500378052173528780792s^{42}-15528080343980828527604766556375005181552902663086802848142696321224s^{41}+238282964650245168300609610231809513127916963399478723451985201685904s^{40}-3430367236060179440070732218664376912627362282540244578551905978189104s^{39}+46365532467244547446360743216142017095989148306459526039623515846047792s^{38}-588755762949838005565806711016983484397068959152714283441733945344802848s^{37}+7027249712549848127431398466867724250461309354203649153646258736184974752s^{36}-78871493856005778995281528812829746466339050516177676306484461418414160416s^{35}+832653557924952811014500182820751911383034261282044436164334245143830456352s^{34}-8269810361173954931078914169946911637391596761131737508844305945597487794144s^{33}+77275880265085319916055821144235805497262635145714014287742658219786671877504s^{32}-679349084345154135379649690958316686561437063246031516423762907925538312289408s^{31}+5617964001822564205130667238529394632220025771132883175722208150517560549227648s^{30}-43690824350734077930796967911338231155740749007361979222513577735261402688590592s^{29}+319422913565064992742490317937286445800967315990365504365040042142459301153592512s^{28}-2194292900885520709718961937109526038033002419198710908780183901772626370705331456s^{27}+14155019838518846812037482168797435038238593984348815057869321629794699529203855616s^{26}-85682417222445926795565973572329420277366975422152054005379234967639143358197815808s^{25}+486245607801686576677241553060465905470056389505993650262344015281342122162160575232s^{24}-2584368483713735021047686463913913640136052794643430856590242035633830225424725988864s^{23}+12848980595866502714148654836356043391589862545760566610324380590058932864681066715648s^{22}-59676137078910398640252148320907121417908906930343892171666836705986652465992239468032s^{21}+258503073948652279571122377723799998795218261647593540516168322657994235939676499460096s^{20}-1042512327688912764291535477319351885866939675291965746264513281156719001472917009644544s^{19}+3906196167448619261818676733246713157402634575850720548507828059644983940218233231610880s^{18}-13566400556764072761657734569104915733372068338209863555592826673337289078412816532291584s^{17}+43555912849470441098868802000969681226084842115362413993508512687803571381115014140829696s^{16}-128873854467322076750495059627569588067583487401622277235861210691551499032718516262586368s^{15}+350170085209767527530821777446058436886835554296012946187110660487759175384250607478042624s^{14}-870180189847506227220609230477835558202475594829635350625252034102067479987046653583601664s^{13}+1968252700727960939216791826879252136521049007411500029477449576057390685915953096594960384s^{12}-4029555319153116978053280766009889585799476763772621260420608643127295976220742724892581888s^{11}+7417299288136909883224183664841980102926026482386873232431581240176756976673835812755652608s^{10}-12177862422215869003301230978568940238954188047906715505518821155614152635401370021975375872s^9+17659998109675690653749221764562831419249106682956363249429630212164222854382762578641272832s^8-22347378432178696652088994803216423167402716129816166966820581562620109814623380610932342784s^7+24296916754024602064906452756050319984681043769067192580115816651405229556935324316920250368s^6-22239046622093583453544338095037087546806112181982198692585543077930687627404701078594387968s^5+16664365087788761186074795638291069350851003090434486799819196634836845588756587148389384192s^4-9816056457754695102925029410241846551198661538424486286292785393106187869550293083972698112s^3+4262107623285338836956260105382884155349919936507235251826717599203646332610595428224729088s^2-1212790846648326411539652251907964256150144830955555521253734442565303245011213397155381248s+169654700822801341540046526070943346738341569829349407278942246963305979508724394755096576=0$
Originally found by Károly Hajba
in September 2015.
Bounded the $s(n^2-n)=n$ conjecture to $n<16$.
Improvement based on the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019 and improved by David Ellsworth in December 2024.
241
$s = \Nn{15.99379865105946}$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n+1) \lt n$ for $n=16$.
Improved by David Ellsworth in November 2024.
Improved again by David Ellsworth in November 2024, by adapting a technique from the $s(240)$ found by Károly Hajba in September 2015.
257
$s = 13 + {5\over 2}\sqrt 2 = \Nn{16.53553390593273}$
Extends the $s(65)$ found by Frits Göbel in early 1979.
This alternative, converting the $s(65)$ augmented by eight "L"s into a primitive packing, found by David Ellsworth in January 2025, using a computer program he wrote.
258
$s = {19\over 2} + 5 \sqrt 2 = \Nn{16.57106781186547}$
Adds an "L" to the $s(227)$ found by David Ellsworth in January 2025 (using a computer program he wrote) which continues the $s(26)$, $s(85)$ series found by Erich Friedman in 1997.
259
$s = {}^{12}🔒 = \Nn{16.60255251726339}$
$64s^6-(4352+288\sqrt{2})s^5+(112386+18636\sqrt{2})s^4-(1295184+466536\sqrt{2})s^3+(4902306+5553822\sqrt{2})s^2+(19879740-30390606\sqrt{2})s-143635023+56063664\sqrt{2}=0$
$4096s^{12}-557056s^{11}+33159424s^{10}-1122522624s^9+23299339236s^8-290070700224s^7+1703658414312s^6+5714085847824s^5-182330704044288s^4+1346739639595488s^3-4105729915848180s^2+1104381069893496s+14344750990000737=0$
Found by David Ellsworth in December 2024, by extending the $s(102)$ found by Károly Hajba in September 2024 and adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Improved by David Ellsworth in December 2024.
Further improvement pending.
260
$s = 11 + 4 \sqrt 2 = \Nn{16.65685424949238}$
Extends the $s(124)$ that continues a pattern found by Frits Göbel in early 1979. This alternative, converting the $s(148)$ augmented by four "L"s into a primitive packing, found by David Ellsworth in December 2024.
261
$s = 16 + {28\over 41} = \Nn{16.68292682926829}$
Adds an "L" to the $s(230)$ found and improved by David Ellsworth in January 2025, which uses a rotational symmetry technique found by David W. Cantrell in January 2025, and the technique from the $s(293)$ improved by David W. Cantrell in January 2025.
262
$s = 16 + {1\over 2}\sqrt 2 = \Nn{16.70710678118654}$
Extends the $s(52)$ found by
Frits Göbel in early 1979.
Explore group
263
$s = {25\over 2} + 3 \sqrt 2 = \Nn{16.74264068711928}$
Found by David Ellsworth in January 2025, based on
the $s(297)$ he found.
264, 265
$s = 9 + {11\over 2}\sqrt 2 = \Nn{16.77817459305202}$
Continues a pattern found by
Frits Göbel in early 1979.
Explore group
266
$s = {}^{32}🔒 = \Nn{16.82306208283780}$
$2401s^{32}-1162084s^{31}+263199482s^{30}-36971795756s^{29}+3593304360857s^{28}-254523535144036s^{27}+13408548240776810s^{26}-519901936408593644s^{25}+13805430523268790240s^{24}-171760035794017558004s^{23}-4426818799267672804198s^{22}+326745541238344524670064s^{21}-10052823560617182774582207s^{20}+181074087488548519346667052s^{19}-1180165534713863709787052484s^{18}-39814767996810201813233201880s^{17}+1608559573416436468095371051805s^{16}-32123383071505854643466513964696s^{15}+412614513240021198536387912093202s^{14}-3166929683248511957762311522115552s^{13}+4864413227687588362660593637481982s^{12}+178773674416065677195226856423404204s^{11}-1078437021027255344046160578887057944s^{10}-30113420634289372798330615891075770272s^9+791230647693671970529285145442783714082s^8-10342818482011428376427887130138935213892s^7+91890490062472897914498395018088233984940s^6-595883639785360944056309451122760067196636s^5+2858083210066295032341927117330041150532457s^4-9961121615142990990156112009477337084333564s^3+23996879793794528190240708983850819688279982s^2-35874639749283350605126520242841070953510704s+25142156270060598069723106950936800398659193=0$
Found and improved by David Ellsworth in November 2024 and in December 2024, based on the $s(53)$ found/improved by David W. Cantrell in September 2002 and December 2024, respectively.
267
$\begin{aligned}s &= 16-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{16.84666719284348}\end{aligned}$
Extends the $s(107)$ found by
Károly Hajba in November 2024.
268
$s = {}^{4}🔒 = \Nn{16.88166675700900}$
$23s^4-1478s^3+35488s^2-377012s+1493621=0$
Found by David Ellsworth
in November 2024, based on the $s(70)$
found by Joe DeVincentis in April 2014.
269
$s = {}^{8}🔒 = \Nn{16.90596764828402}$
$1338s^4-(92136+4180\sqrt{2})s^3+(2378600+206890\sqrt{2})s^2-(27290208+3419060\sqrt{2})s+18867680\sqrt{2}+117431679=0$
$8028s^8-1105632s^7+66453952s^6-2277483296s^5+48690729660s^4-665109430256s^3+5669988144368s^2-27584892248768s+58646728859167=0$
Found by David Ellsworth in November 2024, by extending the $s(88)$ found by Erich Friedman in 1997, and adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Improved by David Ellsworth in January 2025.
Further improvement pending.
270
$s = \Nn{16.95499994790412}$
Found by David Ellsworth in December 2024, by removing 2 squares from the $s(272)$ based on the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
271, 272
$s = {}^{66}🔒 = \Nn{16.98264294789927}$
$1713850177070388831714241s^{66}-1263006180902967521708887682s^{65}+457556854283194653508246334297s^{64}-108626085650108755375383219863016s^{63}+19007543579258154993013605104517540s^{62}-2614249620222774694075991845075567408s^{61}+294320445544799712214422954346585080428s^{60}-27891472736856262183567432641837740604344s^{59}+2270607650860185114208141699776485160991834s^{58}-161270542469096575205017999202879333703379396s^{57}+10115431614750251067098547571363673856667344506s^{56}-565808816641730403858512432034552952494802469080s^{55}+28450183984319404845113532374059123476095944389108s^{54}-1294559647452187836265838445107719695018437168558336s^{53}+53606558683243577767537060160465091965389251586245100s^{52}-2029777274949400008320972989373569605256603237798407896s^{51}+70566124724622022053181378677212143452798761833741049255s^{50}-2260494118677599144499162628207117877734197301937499627454s^{49}+66928391584632217881721045867244937443110647589022537806159s^{48}-1836487293516144718606242405600092685339691196348395055392352s^{47}+46812433957159983606351028994995744914840143394813517611414488s^{46}-1110784325503673809119053441923324973802716059719307513683034144s^{45}+24580067642598124655873296327812538035270416503768622579007969464s^{44}-508058601195291180778593698054098715271789503226858668644025004528s^{43}+9822636844453945473008973446300398331806350662367666559112039098828s^{42}-177850157754818541348101942178417197242325906137100400008326274953784s^{41}+3018908592957234274621453725329581711584511453209430776141062678498508s^{40}-48084823616106717807937793455351705691203975334728299955852985477677936s^{39}+719219081331785321193765357384648827974404178174478952936526177053613640s^{38}-10108484874977669848442591400277761623042675059618304149046369427119689280s^{37}+133569141781888512562398911029947506237982901071358900752904689441632946104s^{36}-1659947237436600687047903428474338638444186680114020006663182877218281431536s^{35}+19407721788005616167841988847134491979124178917074454947061592816288392155103s^{34}-213512563646549268743663877471869657452486579360256542887702449445267477625758s^{33}+2210398398130635465592076383059864672001655782088693299537565902349619712869319s^{32}-21532723313632709226861394724764610661805015535470154323486739829302573530834280s^{31}+197353488882411915369080553242112202824443643000941071851504553934557696342479860s^{30}-1701361607748327036898500039362885770951773534397536787240248737455726000275481776s^{29}+13790944656640719079864365329065256075236685711621827154509298129212864382169075708s^{28}-105057191253365590055038856706594446361850240187411125808555225806066647805699033432s^{27}+751670513866794149869446078936059679095796755093931999846310783101254874128343844858s^{26}-5047524051032968924904168688747563678450991211248789313855857351391502072525065835844s^{25}+31783178768787466515646386129275724683780750656572320239619905303340921614133673928026s^{24}-187472727052955447696160431653518028007026362437178333588607245141115925448495562239416s^{23}+1034626229621705034003065868277725184320524563387572538925459708898162378122695884371524s^{22}-5335045699393078934531764329130636548870723955424210927081461414593591826916976872572736s^{21}+25663737575409659302307158237342619977251070651367656518135022159442843909863332058148316s^{20}-114960649295229282113311852806854796874367423592350138710159368893751322473994310949804856s^{19}+478559753673706084748519492363770439999236334487313334845093021068854656585876271428583577s^{18}-1846990700033721376192730686948971722144142619617398680320187325046521383970748784637316834s^{17}+6591326456994561029578640106261265890282504157582308701300923406400578110653255439076001265s^{16}-21683429758094255971784837646080964873309338986609460623022430958540451763873664347265934416s^{15}+65523328529399452484083881848848270952170741513603155833639414320711386575541360843917428752s^{14}-181134395473848110073827498749917703659200447864218040383057641661298047344658812692009492864s^{13}+455904206381436590247441608773490326831830404955375957062019998388748856279690733152466535776s^{12}-1038920953582196812525580854613110667241159361649685005599580196703726210365369892962261589248s^{11}+2129321555265993058808909413739742507947670301867618671926711075672095702442458870941704884224s^{10}-3893858393082444876460677140167356642408686917745920213899715750218779803640863536637271852544s^9+6291634657144980094912045673643258772753140381469197256391134629731334665393467914754165776640s^8-8874016200990822056453884612581892989147523385532622735339375442773419838200440294959402524672s^7+10757961171340950446264308166124992160748835994596014475038180685505768156638580567927181983744s^6-10983792379878536403573363876927768332193956898933674258258398402852665358347513825072163651584s^5+9184651747912342152838771898129517117129618056068600195075502388329608767166607606008661409792s^4-6040013853226973048374847053430821660325834738741764444484502435456762249243146187494035292160s^3+2929211695042187145319679151141672508929021947850940754766129715898448812797031858745271910400s^2-931421152741943199632685430128893095342601525238972381704891083745166622082394765118220009472s+145672913717139262806845359990077919840464280273804604430606108170184726692142354311210336256=0$
Originally found by Lars Cleemann between 1991 and 1998.
Bounded the $s(n^2-n)=n$ conjecture to $n<17$.
Improvement based on the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Improved by David Ellsworth in December 2024.
273
$s = {}^{5}🔒 = \Nn{16.99295005377111}$
$10s^5-412s^4+7204s^3-73814s^2+416937s-935217=0$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n+1) \lt n$ for $n=17$.
Improved by David Ellsworth in November 2024, by adapting a technique from the $s(240)$ found by Károly Hajba in September 2015.
Improved by David Ellsworth
in December 2024.
290, 291
$s = 14 + {5\over 2}\sqrt 2 = \Nn{17.53553390593273}$
Combines two copies of the $s(65)$ that continues a pattern found by Frits Göbel in early 1979.
This is the first $s(n^2+2)$ that has the same side length as the best known $s(n^2+1)$, and is likely an irreducible primitive packing.
292
$s = {}^{12}🔒 = \Nn{17.60255251726339}$
$64s^6-(4736+288\sqrt{2})s^5+(135106+20076\sqrt{2})s^4-(1789528+543960\sqrt{2})s^3+(9506654+7068126\sqrt{2})s^2-(42973842\sqrt{2}-5717888)s-157200471+92493552\sqrt{2}=0$
$4096s^{12}-606208s^{11}+39557376s^{10}-1485656064s^9+34988158692s^8-521042431968s^7+4488051338520s^6-12054166969344s^5-173414241628704s^4+2087574544612928s^3-9264733241063660s^2+14101503785216640s+7601873759468433=0$
Adds an "L" to the $s(259)$ found and improved by David Ellsworth in December 2024 extending the $s(102)$ found by Károly Hajba in September 2024 and adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Further improvement pending.
293
$s = 17 + {26\over 41} = \Nn{17.63414634146341}$
Found by David Ellsworth in December 2024.
Improved by David W. Cantrell
in January 2025.
This is the first record-setting packing found with a rational side length, thanks to the Pythagorean triple $\{20, 21, 29\}$ determining its tilt angle. It was foreshadowed just 3 days earlier by David Ellsworth finding an $s(104)$ with rational side length.
294
$s = 12 + 4 \sqrt 2 = \Nn{17.65685424949238}$
Extends the $s(148)$ that continues a pattern found by Frits Göbel in early 1979.
This is likely an irreducible primitive packing.
295, 296
$s = 17 + {1\over 2}\sqrt 2 = \Nn{17.70710678118654}$
Extends the $s(52)$ found by
Frits Göbel in early 1979.
Explore group
297
$s = {27\over 2} + 3 \sqrt 2 = \Nn{17.74264068711928}$
Found by David Ellsworth
in January 2025.
Extends the $s(85)$ found by Erich Friedman in 1997.
Improvement pending.
298
$s = 10 + {11\over 2}\sqrt 2 = \Nn{17.77817459305202}$
Adds an "L" to the $s(265)$ that continues a pattern found by Frits Göbel in early 1979.
299
$s = {33\over 2} + {1\over 2}\sqrt 7 = \Nn{17.82287565553229}$
Extends the $s(86)$ found by
Erich Friedman in 1997.
300
$s = {}^{40}🔒 = \Nn{17.82412338847854}$
$6765201s^{40}-4468746888s^{39}+1431055332684s^{38}-295991953326108s^{37}+44445603736025352s^{36}-5163220519287531048s^{35}+482878985597614632366s^{34}-37353497397438488815260s^{33}+2436831832955268577880066s^{32}-136013384348919954233715980s^{31}+6566863009941672632548971230s^{30}-276576685299846988827282188436s^{29}+10227647719730769459718165452749s^{28}-333724070621087598910090605855348s^{27}+9643524705514237911246918698322714s^{26}-247404094366714093683897197709917204s^{25}+5643119674396861410915876260884036989s^{24}-114477712300768469604781612467804162660s^{23}+2063707000134940629708077092426963028132s^{22}-32985932723685807897922726019150035754248s^{21}+465583759504110006491405487605416144111553s^{20}-5764247971291523576624473899585978835759824s^{19}+61914526137823063569936032519366359362310830s^{18}-566147126795280425805190165571461577530656336s^{17}+4248816809091624736372842695988910574840203032s^{16}-23948399126640716409385371068745856498999580804s^{15}+69800847299887214247772249238723707512987918380s^{14}+394782152673201159198736947784032596106016880136s^{13}-7828936745814934561867828618761111333890039348250s^{12}+67387777391090638044324068706351575262477706086740s^{11}-381562083381768156325668597682082463480074995868360s^{10}+1336650893593888696724108652818662244745263296865588s^9-614601349329475752238933120184261110575651083354603s^8-28305247058854021955410328849850580588215220244356248s^7+224981301926271768349655651977718726650378985178322646s^6-1055948455747248748890178989019952798681204013121554736s^5+3473810396018259264801879879806884179343515176608754025s^4-8177313821139351944992551555783250987371382184946020000s^3+13278518858240943321872383441026571982205038824476375000s^2-13426397690512873291977824583646716429736153596450000000s+6401405918124622124119207874790462553248468260156250000=0$
Found by David Ellsworth in January 2025, extending the $s(128)$ and $s(205)$ he found, based on the $s(69)$ found by Maurizio Morandi in June 2010 and using the technique from the $s(53)$ improved by David W. Cantrell in December 2024.
Improved by David Ellsworth in January 2025, using the technique from the $s(128)$ improved by David W. Cantrell and David Ellsworth in January 2025.
301
$s ≈ 17.87595$
ROUGH DRAFT, UNFINISHED.
There are some micro-overlaps, and the size is approximate.
Found by David Ellsworth in January 2025, including adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Extends the $s(11)$ found by Walter Trump in 1979.
Planned to be finished using a computer program yet to be written.
302
$s = {}^{4}🔒 = \Nn{17.88674602860566}$
$s^4-66s^3+1631s^2-17882s+73369=0$
Found by David Ellsworth in December 2024 (including re-adapting the techniques from the $s(130)$ and $s(129)$/$s(206)$ he improved), by extending the $s(88)$ found by Erich Friedman in 1997, and adapting and extending the $s(37)$ improvement found by David W. Cantrell in September 2002.
Further improvement pending.
303
$s = \Nn{17.95499994790412}$
Adds an "L" to the $s(270)$ found by David Ellsworth in December 2024 by removing 2 squares from the $s(272)$ based on the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Improvement pending.
304, 305
$s = {}^{66}🔒 = \Nn{17.98264294789927}$
$1713850177070388831714241s^{66}-1376120292589613184602027588s^{65}+543328464671703526463351080572s^{64}-140615202964614126638004715304744s^{63}+26829824909531386795122409867553580s^{62}-4024798138448651535381545184824888272s^{61}+494347896425777838963701966211841094208s^{60}-51121940635378797836287264998905061474368s^{59}+4542640548237608922133896403568612165085598s^{58}-352254670142390357290980730103805413351130752s^{57}+24128195020496036167298549899657263698522042112s^{56}-1474179150135552799244494323027883402848618735872s^{55}+80985138091690144145355687368928269630738567897484s^{54}-4026981590548665867874487939909468960023196494310192s^{53}+182267200638373063745230585503289601386963559885303288s^{52}-7545112281899612510165763435068918446604608201263951392s^{51}+286834925628382194495660904989654559193002073657037773457s^{50}-10049565828457746164374711704047033495168594843678677316244s^{49}+325499384321545190494163072511087433536420191967280816128740s^{48}-9772581289162465359405567577566877285512203772853480321141608s^{47}+272613842003155736225570794797609838660943262882211714349718640s^{46}-7080493440341343834411011123586959947754962517882504495538745472s^{45}+171531474757097410249414924053861168536230110955969576731895246920s^{44}-3882214349259127305137310567801724956533274558662561393205688835264s^{43}+82200585145535219135810965054893272960077010195154270039200863749688s^{42}-1630258956865275295872337774636555763342793225726628107661326519656232s^{41}+30316602368448156150200109954721666553328238001770420734846520369155456s^{40}-529100170410512162733967021266362515006823176250235236208800862485778096s^{39}+8672812114309507583523769261574291633843885317390927008959165725551281104s^{38}-133604712858611307770326141488963196041431007457229154153884192004409913632s^{37}+1935287963280671887853021351300551803147977152959329110457304496325932053152s^{36}-26369568984008683547580706652585786757749645465630170649594991804279169086240s^{35}+338078080839919096674573098826845065127283338268117746286742900602332756029664s^{34}-4079087539789791575560050969074532457336361603032072778873961672817546356973536s^{33}+46320137173197740753817591401820230866740557176026937020010829487638066263326912s^{32}-495016582000784435413106529483575249511354685535037968758201031333595965076898944s^{31}+4977924700882517458359048737510231963396017072296127685672337183902430724422017920s^{30}-47091620430092681534926593487054276521041403353074011105123802349042439773476528896s^{29}+418933385016745211947222808620255441114380728391860527906265870911858034743285241024s^{28}-3503006886259715819412667649026476439477808686694534314551125185742311900556660285184s^{27}+27514857713737746963130211058679811954406316972651571150660641469472381982354794337536s^{26}-202863150403975114963635408680873544302505729554954613863476699751624919208660885852672s^{25}+1402709995829726084689277339133025652193049640394127529908133471556083038663917684053760s^{24}-9086899324658301009708796843048145448075340279291908987325520939240439100244688624002560s^{23}+55084584772479049650257779664648209447949122588055864610352900437017173418201795772243456s^{22}-312044443572486729316802467807724154381090514130070572414647103019127107672636876664290816s^{21}+1649277211093445693448208153574041967523560611293184471904202237863220666468961287187338240s^{20}-8118640175848026108993106281833917188694597678315910558063402475202051432613391955855641600s^{19}+37144696681372172439650831549400091380866366509181988973869339080686979212069965990823787520s^{18}-157586730651964411252322536014125965772239288740159848629723235086393681235894364201883713536s^{17}+618288927781270563652014578600966260282549165002258008110053288490780862741600781299207507968s^{16}-2236565450657513656482121703891867471823841248110283410884806433679812901105466655396471359488s^{15}+7432893071780345274664428029482179027513487039424442339358853964292563899965270358695764254720s^{14}-22601990380304840114232955816799025270171303224566573276800545756165651975962250575537444556800s^{13}+62586563073321683259360467444111903207735246150889958394168784027878147573962885126129204400128s^{12}-156939492712340281577011813896174237722581326407907915345344934496499932397944939733935650037760s^{11}+354010766890679907447918498245949709618769477101420391146521973712262419115277237946938967687168s^{10}-712634824084661388879909049188671154939579018589650125862407413872672097371655108017943957979136s^9+1267801797993963237889272907735359448760706717740504858777243285714820373043141123507911382614016s^8-1969253813038576659750419438966806080674935670243968164986133969042920568174489538190545489821696s^7+2629669713639732187218315553568862244094923460774770810827814913228347803950554622508321583857664s^6-2958102093115310789329465782332338661036926390430105544964292955398971352437590664062597062066176s^5+2725943978218028478771349856122398946519913316361589789870009883220796007931840915720001460109312s^4-1976029693513971350410426740321961150195134039008222100615918618954101379818384436418594287386624s^3+1056622505101796269393565035252896788783387631720299838353179437245837174921473700994668720816128s^2-370548971618299668657098004068243761126772930024009924355548824210287559967194161234577752850432s+63933786130214659824460347409445038461589338575833265414758677167009207593207893617517426376704=0$
Adds an "L" to the $s(272)$ originally found by Lars Cleemann between 1991 and 1998, and improved based on the $s(210)$ found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019, and then improved by David Ellsworth in December 2024.
Improvement pending.
306, 307
$s = {}^{5}🔒 = \Nn{17.99295005377111}$
$10s^5-462s^4+8952s^3-97998s^2+587875s-1433594=0$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n+1) \lt n$ for $n=18$.
Improved by David Ellsworth in November 2024, by adapting a technique from the $s(240)$ found by Károly Hajba in September 2015.
Improved by David Ellsworth and Károly Hajba in December 2024.
626
$s = {27\over 2} + {17\over 2}\sqrt 2 = \Nn{25.52081528017130}$
Found by David Ellsworth in January 2025, using a computer program he wrote.
Continues the $s(26)$, $s(85)$ series found by Erich Friedman in 1997.
1453
$s = {}^{62}🔒 = \Nn{38.62811880681648}$
$14057529258938368s^{62}-16144332707905069056s^{61}+4126908192850362814464s^{60}-114048398674982625548288s^{59}-52812457965071023131108736s^{58}+3123757315293716451259828736s^{57}+395914808055074354598655919872s^{56}-25655581919693099977675345288928s^{55}-2342138210856135887439487492867911s^{54}+130614558810433031715467943008591574s^{53}+9582864850318173225492635922395708673s^{52}-400208116572281336325713849443556893320s^{51}-19278076716042492939533753042312495338866s^{50}+155639463279626013183461774071774170428332s^{49}-22682951311029575205053841559156685327668694s^{48}+4481148848166715045575771589473008085986195508s^{47}+339908983431725009307125627170625587965064721641s^{46}-21836898654237099253200945764943586503872843975282s^{45}-1733607911337837421674786648905767199367409287630091s^{44}+65577710215566787889219323647412546993333020408126340s^{43}+6575020048126915878362134657166967660162756051625767082s^{42}-189919577494849234065185357103567819076518682190184112052s^{41}-18786350786500438406048733796469585237988085985102482357718s^{40}+577860661366947067720361099304613584329095511092740182446472s^{39}+38560113418007612161265092791117116831719699647020808115500290s^{38}-1509778230715804736044763817572589857709434302255396595167198724s^{37}-53691061034745282967831861830008396823011099992793727965608258808s^{36}+2981962876446433126961251394849875590796626863624033992874793421944s^{35}+42946650830136738474116255305854244294539529383743966678795990596614s^{34}-4305207796096057376307480627217176796778265873544269426186757603548600s^{33}+847984922128592247433827812551628892707545393068784907659271398018534s^{32}+4472381287851045675238737659317271974769806475510817553632400089733850984s^{31}-54024990039620679626108375115227820285123938297900797999074130495666406019s^{30}-3193681274533744120801748520386719645289103551859824962842504952991692546322s^{29}+80917624554132100648898693934599191301469617060018407110555412748900902279137s^{28}+1304198975666560613729453090086267184069864122474264569263280366382519245805416s^{27}-69287844982281380279236661128207636125050487817999177431635188503445709265641520s^{26}+68996038980379916684164096301103219794837210102541034822026299278384997449217692s^{25}+37867353091715263710849645403683336968945849908862704327076999773411927727397591158s^{24}-509915867359895925422359611092573447179100380327510553509286921447269433977643865508s^{23}-11683211944540307067146210206410836570815757425407896484941417171550422135445264863064s^{22}+368827309307507537219147042570180764187110525313995270219698774784754026548995528531660s^{21}+185724218102520508187862654139453737315454289351002925523872644784163316111213102216680s^{20}-136830853415425229763805282272266732584005552776626700663010740658119017917916268390287556s^{19}+1547848085441210529100258771205521079963814294017010309696882835752388457232036205727597039s^{18}+21914646855044175829856156796290016500376182741174287509692603427850292470216693384112852346s^{17}-642013029716693421472434447425253341135587415866685406570588222982341705951674252923582041795s^{16}+2430889433553432642414397495285152141337085610751304554696488205951846350369700845506363822732s^{15}+92064055623459052561344041325712052237179537558423822365643138628075940784085506754380742599882s^{14}-1436706418001841492955377568562805242864121009591891493210031742373552154899343547291316137557908s^{13}+4993033976064547466127206717163229131723638752386640105025500484333917749253388698555336828511242s^{12}+96031442149539838915450548148896134358831811633526781603086096632377321989140256994318562339417100s^{11}-1733048273294974846832844855482169873194419439395735002201619922770668316679232678681162587124552260s^{10}+15770820434262396395851079386321352123276831933189279029980483538751595421431357063482729417803984580s^9-97278075882540976992627527924963133628721769353102052957581703613903792317825205295534444102067365466s^8+463911493164648755268346400550691149594031202366680322675918086734493393306092169030187572680145939860s^7-2090656957701197789966258215824701051567847005345271229373699635289968414443018273787166250135969126875s^6+9442609168926974312630527590013048294221786119093364465176592130998930554703600095438263441086719047690s^5-39415054084443633566696762271703899900300756149352769966371040153484310767724118279361770290440553743801s^4+156031640636394568927996748174625458407871841132601217731374082667948155491740616882144255248507174778908s^3-496791378964789968261213028274305514593846288225401594028735451597189456726232160601931666507937957449457s^2+1305836137839118688998085219859481571482741756188636214545184411851028521415422566589743034644001594557270s-3410485277066865875954124600874980278861680322256497845789816598547317193010330644022088333312859866677675=0$
Found by David Ellsworth
in December 2024.
Extends the $s(17)$ found by John Bidwell in 1998, and the $s(83)$ found by Károly Hajba in September 2024 and improved by David W. Cantrell in November 2024.
See also
$s(260)$,
$s(446)$,
$s(791)$,
and
$s(1097)$, none of which are optimal.
1765
$s = {}^{4}🔒 = \Nn{42.48797851186022}$
$2s^4-212s^3+8129s^2-148140s+1362276=0$
Found by Károly Hajba
in November 2024.
Bounds $\{s(n^2\!+\!1)\} \ge {1\over 2}$ to $n \lt 42$.
Beats the $s(1765)$ Göbel square.
Improved by David Ellsworth
in November 2024.
1850
$s = {}^{4}🔒 = \Nn{43.48878088476276}$
$2s^4-224s^3+9311s^2-185004s+1705932=0$
Found by Michael J. Kearney
and Peter Shiu in June 2001.
Bounded $\{s(n^2\!+\!1)\} \ge {1\over 2}$ to $n \lt 43$.
2043
$s = {}^{12}🔒 = \Nn{45.69644276992823}$
$4s^{12}-1608s^{11}+293084s^{10}-31920420s^9+2301941449s^8-114905182392s^7+4022452365218s^6-97595016541596s^5+1574653827588509s^4-15396232508703888s^3+72639007870740216s^2-58090491554723760s+46014771089277232=0$
Found by David Ellsworth
in December 2024.
Beats the $s(2043)$ Göbel strip.
