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.
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 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 $s(66)$ found by
Evert Stenlund in 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.
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.
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.
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. 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 1980.
67
$s = 8 + {1\over 2}\sqrt 2 = \Nn{8.70710678118654}$
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.
Improves upon the earlier $s(83)$ that
added an "L" to the $s(66)$ found by
Evert Stenlund in 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 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.
87
$s = {}^{6}🔒 = \Nn{9.85197533993158}$
$9s^6-282s^5+2045s^4+5870s^3-89663s^2+554156s-3691596=0$
Found by Evert Stenlund
in 1980.
Improved by David W. Cantrell
in August 2002.
88
$s = {}^{8}🔒 = \Nn{9.888160100452437}$
$4898s^4-(186856+8508\sqrt{2})s^3+(2676964+234258\sqrt{2})s^2-(17078860+2159580\sqrt{2})s+40971345+6672600\sqrt{2}=0$
$9796s^8-747424s^7+24905648s^6-473558928s^5+5621403852s^4-42670393040s^3+202313221240s^2-547916568600s+649082862225=0$
Found by Erich Friedman in 1997.
Improved by David Ellsworth in November 2024, by adapting and extending the technique from the $s(37)$ found by David W. Cantrell
in September 2002.
89
$s = 5 + {7\over 2}\sqrt 2 = \Nn{9.94974746830583}$
Found by Evert Stenlund in 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
$s = {}^{8}🔒 = \Nn{10.61350257784129}$
$392s^4-(33432+13888\sqrt2)s^3+(706840+371152\sqrt2)s^2-(5716880+3287856\sqrt2)s+16051345+9689424\sqrt2=0$
$3136s^8-534912s^7+26247104s^6-635223424s^5+8904202320s^4-76237792816s^3+395289628464s^2-1144847001376s+1426036763377=0$
Found by Károly Hajba
in September 2024.
Improved by David W. Cantrell and
David Ellsworth in November 2024,
by extending the $s(37)$ found by
David W. Cantrell in September 2002.
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
$s = {}^{144}🔒 = \Nn{10.92591939016138}$
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Found by Károly Hajba
in October 2024.
Improved by David Ellsworth
in November 2024.
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
$s = {}^{12}🔒 = \Nn{11.60304782603491}$
$16s^6-(808+96\sqrt{2})s^5+(17055+4248\sqrt{2})s^4-(193408+74592\sqrt{2})s^3+(1249979+648684\sqrt{2})s^2-(4396854+2788260\sqrt{2})s+6628736+4726152\sqrt{2}=0$
$256s^{12}-25856s^{11}+1180192s^{10}-32118704s^9+578685345s^8-7241250352s^7+64139810474s^6-401461100180s^5+1735510964089s^4-4911120357412s^3+8091982299332s^2-5580206491008s-732884496512=0$
Found by David Ellsworth in December 2024, by extending the $s(102)$ found by Károly Hajba in September 2024 and adapting and extending the technique from the $s(37)$ found by David W. Cantrell in September 2002 and the $s(130)$ improvement by David W. Cantrell in November 2024.
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, 127
$s = {21\over 2} + {1\over 2}\sqrt 7 = \Nn{11.82287565553229}$
Extends the $s(86)$ found by
Erich Friedman in 1997.
128
$s = {}^{4}🔒 = \Nn{11.82629667551039}$
$2s^4-82s^3+1248s^2-8354s+20759=0$
Found by David Ellsworth
in November 2024, based on the $s(69)$
found by Maurizio Morandi in June 2010.
129, 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 by
David W. Cantrell in September 2002.
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
$s = {}^{12}🔒 = \Nn{12.60304782603491}$
$16s^6-(904+96\sqrt{2})s^5+(21335+4728\sqrt{2})s^4-(270028+92544\sqrt{2})s^3+(1940853+898908\sqrt{2})s^2-(7549392+4326876\sqrt{2})s+12486856+8242032\sqrt{2}=0$
$256s^{12}-28928s^{11}+1481504s^{10}-45399024s^9+925255281s^8-13177364728s^7+133989746070s^6-975129113048s^5+5003168814793s^4-17439265483424s^3+38384597583024s^2-45887339774976s+20059389786688=0$
Adds an "L" to the $s(123)$ found by
David Ellsworth in December 2024.
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
$s = {23\over 2} + {1\over 2}\sqrt 7 = \Nn{12.82287565553229}$
Extends and adds an "L" to the $s(86)$ found by Erich Friedman in 1997.
151
$s = {{3154 - 26\sqrt 6}\over 241} = \Nn{12.82287662525990}$
Found by David Ellsworth
in November 2024.
152, 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
$s = \Nn{12.97350182495032}$
Adds an "L" to the $s(131)$ found by
David Ellsworth in November 2024.
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
$s = \Nn{12.99404229036268}$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n)<n$ for $n=13$.
170
$s = 10 + {5\over 2}\sqrt 2 = \Nn{13.53553390593273}$
Adds five "L"s to the $s(65)$ found by Frits Göbel in early 1979.
171
$s = {}^{8}🔒 = \Nn{13.59861960924436}$
$6s^4-(376+64\sqrt2)s^3+(8190+2194\sqrt2)s^2-(75556+24966\sqrt2)s+253307+94710\sqrt2=0$
$36s^8-4512s^7+231464s^6-6503888s^5+110915328s^4-1184746768s^3+7780100524s^2-28819607944s+46224468049=0$
Combines two copies of the $s(37)$ found by David W. Cantrell in September 2002.
172
$s = {}^{8}🔒 = \Nn{13.61970688383567}$
$562s^4-(166824+99232\sqrt2)s^3+(5529694+3579002\sqrt2)s^2-(64343912+42896630\sqrt2)s+252737371+171045700\sqrt2=0$
$1124s^8-667296s^7+51073464s^6-1767572224s^5+34464149128s^4-405438721504s^3+2869488680332s^2-11299238055184s+19085109334561=0$
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.
173
$s = 8 + 4 \sqrt 2 = \Nn{13.65685424949238}$
Adds an "L" to the $s(148)$ which 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
$s = {}^{8}🔒 = \Nn{13.81880916998841}$
$5184s^8-487872s^7+19993424s^6-465928808s^5+6752167261s^4-62298031950s^3+357291488120s^2-1164321164284s+1650202854142=0$
Combines two copies of the $s(39)$ found by David W. Cantrell
in August 2002.
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 in 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
$s = {}^{4}🔒 = \Nn{13.93786550630255}$
$s^4-44s^3+725s^2-5272s+14036=0$
Combines two copies of the $s(41)$ found by Joe DeVincentis in April 2014, which fits an $s(n^2\!-\!n\!-\!1)$ pattern.
180
$s = {}^{4}🔒 = \Nn{13.97970624703929}$
$s^4-32s^3+389s^2-2062s+2036=0$
Adds an "L" to $s(155)$, which continues the $s(n^2\!-\!n\!-\!1)$ pattern found by Joe DeVincentis in April 2014.
181, 182
$s = \Nn{13.98318264138415}$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n)<n$ for $n=14$.
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 technique from the $s(37)$ found by
David W. Cantrell in September 2002.
199
$s = {}^{8}🔒 = \Nn{14.61970688383567}$
$562s^4-(169072+99232\sqrt2)s^3+(6033538+3876698\sqrt2)s^2-(75906020+50352330\sqrt2)s+322778363+217620564\sqrt2=0$
$1124s^8-676288s^7+55776008s^6-2088089168s^5+44091546248s^4-562015927840s^3+4311045608500s^2-18401537158840s+33695558254817=0$
Adds an "L" to $s(172)$, which extends the $s(102)$ found by Károly Hajba in September 2024 and improved by David W. Cantrell and David Ellsworth in November 2024.
200
$s = 9 + 4 \sqrt 2 = \Nn{14.65685424949238}$
Adds two "L"s to the $s(148)$ which 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, 204
$s = {27\over 2} + {1\over 2}\sqrt 7 = \Nn{14.82287565553229}$
Extends the $s(86)$ found by
Erich Friedman in 1997.
205
$s = 12 + 2 \sqrt 2 = \Nn{14.82842712474619}$
Found by Károly Hajba in November 2024, by extending the $s(28)$ and $s(40)$ found by Frits Göbel in early 1979.
206, 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 technique from the $s(37)$ found by David W. Cantrell in September 2002 and the $s(130)$ improvement by David W. Cantrell in November 2024.
208, 209, 210
$s = \Nn{14.98318264138415}$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n)<n$ for $n=15$.
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
$s = {}^{8}🔒 = \Nn{15.59153604974246}$
$7889s^4-(527752+66432\sqrt{2})s^3+(12889211+2788168\sqrt{2})s^2-(137062792+38929560\sqrt{2})s+537849584+180940800\sqrt{2}=0$
$7889s^8-1055504s^7+59964726s^6-1904714944s^5+37190498217s^4-458703920560s^3+3498811994208s^2-15117574901504s+28368998453504=0$
Adds an "L" to $s(198)$, which extends the $s(102)$ found by Károly Hajba in September 2024, and was improved by David Ellsworth in December 2024.
228
$s = {}^{8}🔒 = \Nn{15.61205342220945}$
$1568s^4-(152672+46784\sqrt2)s^3+(4577260+1970944\sqrt2)s^2-(55519100+27618080\sqrt2)s+238931729+128802168\sqrt2=0$
$50176s^8-9771008s^7+679296768s^6-24549228544s^5+524805011408s^4-6925893926752s^3+55687992448488s^2-251050505939960s+487926003263857=0$
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.
229
$s = 10 + 4 \sqrt 2 = \Nn{15.65685424949238}$
Adds three "L"s to the $s(148)$ which continues a pattern found by Frits Göbel in early 1979.
230, 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
$s = 13 + 2 \sqrt 2 = \Nn{15.82842712474619}$
Extends the $s(40)$ found by
Frits Göbel in early 1979.
235
$\begin{aligned}s &= 15-{1\over 2}\sqrt 2+\sqrt{1+\sqrt 2} \\ &= \Nn{15.84666719284348}\end{aligned}$
Combines two copies of the $s(54)$ found by Joe DeVincentis in April 2014.
236
$s = {}^{8}🔒 = \Nn{15.87893084682989}$
$16s^4-(752+104\sqrt2)s^3+(13186+3952\sqrt2)s^2-(103642+48724\sqrt2)s+315949+192660\sqrt2=0$
$256s^8-24064s^7+965824s^6-21504256s^5+288352740s^4-2358058568s^3+11280245560s^2-27942509156s+25588019401=0$
Found by David Ellsworth
in November 2024, by extending the $s(88)$ found by Erich Friedman in 1997 and adapting the $s(102)$ improvement by David W. Cantrell in November 2024.
237, 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
$s = \Nn{15.98318264138415}$
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.
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
$s = 13 + {5\over 2}\sqrt 2 = \Nn{16.53553390593273}$
Adds eight "L"s to the $s(65)$ found by Frits Göbel in early 1979.
258
$s = {}^{8}🔒 = \Nn{16.58884686366654}$
$1864s^4-(130112+14184\sqrt{2})s^3+(3340332+638892\sqrt{2})s^2-(37526084+9576432\sqrt{2})s+156111517+47789280\sqrt{2}=0$
$14912s^8-2081792s^7+124375616s^6-4175466176s^5+86460492992s^4-1133641742816s^3+9208543852232s^2-42428775277352s+84992168129633=0$
Found by David Ellsworth in November 2024, by extending the $s(102)$ found by Károly Hajba in September 2024 and adapting and extending the technique from the $s(37)$ found by
David W. Cantrell in September 2002.
259
$s = {}^{8}🔒 = \Nn{16.60651495297546}$
$17714s^4-(1505768+323208\sqrt{2})s^3+(43929590+14599998\sqrt{2})s^2-(540077424+219437586\sqrt{2})s+2402899929+1097885178\sqrt{2}=0$
$35428s^8-6023072s^7+408123288s^6-14966022368s^5+330968229088s^4-4567307767008s^3+38656232063604s^2-184242397528080s+379725001766289=0$
Found by David Ellsworth in December 2024, by extending the $s(102)$ found by Károly Hajba in September 2024 and adapting and extending the technique from the $s(37)$ found by
David W. Cantrell in September 2002.
260
$s = 11 + 4 \sqrt 2 = \Nn{16.65685424949238}$
Adds four "L"s to the $s(148)$ which continues a pattern found by Frits Göbel in early 1979.
261, 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, 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
$s = {}^{8}🔒 = \Nn{16.90810332579893}$
$11534s^4-(776040+34380\sqrt{2})s^3+(19549332+1653822\sqrt{2})s^2-(218566796+26528724\sqrt{2})s+915240215+141924780\sqrt{2}=0$
$23068s^8-3104160s^7+182215728s^6-6096166544s^5+127172500628s^4-1694326878096s^3+14081791966552s^2-66762945978200s+138265886040775=0$
Found by David Ellsworth
in November 2024, by extending the $s(88)$ found by Erich Friedman in 1997, and adapting and extending the technique from the $s(37)$ found by David W. Cantrell in September 2002.
270, 271, 272
$s = \Nn{16.98318264138415}$
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.
273
$s = \Nn{16.99379865105946}$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n+1) \lt n$ for $n=17$.
Improved by David Ellsworth in November 2024, by adapting a technique from the $s(240)$ found by Károly Hajba in September 2015.
307
$s = \Nn{17.99379865105946}$
Found by M.Z. Arslanov, S.A. Mustafin, and Z.K. Shangitbayev in March 2019.
Shows $s(n^2-n+1) \lt n$ for $n=18$.
Improved by David Ellsworth in November 2024, by adapting a technique from the $s(240)$ found by Károly Hajba in September 2015.
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 $s(83)$ improved by David W. Cantrell in November 2024.
See also $s(446)$, $s(682)$, and $s(968)$, 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.
