The following pictures show $n$ unit squares packed inside the smallest known square (of side length $s$). For the $n$ not shown, the closest packing shown to the right of it is the best known packing (to have the appropriate number of arbitrarily chosen squares removed). Where a polynomial root is known for $s$ of degree $3$ or higher (and has no concise closed-form expression), a 🔒 icon is shown; click this to see the polynomial root form of $s$.
See also flat grid view and older records and/or alternative packings. For more information on each packing, view its SVG's source code. Zoom:|
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. | |||||||||||||||||||||||||||||||||
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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. | ||||||||||||||||||||||||||||||||||
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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 | ||||||||||||||||||||||||||||||||||
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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. |
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 | |||||||||||||||||||||||||||||||||
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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. |
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. |
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 | |||||||||||||||||||||||||||||||
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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 |
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. |
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 | ||||||||||||||||||||||||||||||||
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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 |
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.
$s = {}^{20}🔒 = \Nn{9.888153053758572}$
$3528s^{10}-(300552+15456\sqrt{2})s^9+(11614660+1180832\sqrt{2})s^8-(268405824+40209136\sqrt{2})s^7+(4111948776+801750848\sqrt{2})s^6-(43682208312+10328732976\sqrt{2})s^5+(326223055436+89277369408\sqrt{2})s^4-(1692962073984+518553084040\sqrt{2})s^3+(5849274524474+1954912407552\sqrt{2})s^2-(12163170266098+4347871933856\sqrt{2})s+11572065260145+4353477802040\sqrt{2}=0$
$254016s^{20}-43279488s^{19}+3506260608s^{18}-179642577984s^{17}+6530192527760s^{16}-179088304328704s^{15}+3846118270819200s^{14}-66261902137415296s^{13}+930479746642904384s^{12}-10759858027891736896s^{11}+103070340120029179008s^{10}-819709665351861223904s^9+5405590814889373243192s^8-29412949608198679086720s^7+130831566348158107359392s^6-468664620024162429231904s^5+1321046745485882223459068s^4-2825402176181244872057384s^3+4315682289270565775115128s^2-4199847844458434080013540s+1959329723251932809573425=0$
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 by David Ellsworth in January 2025. |
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 | ||||||||||||||||||||||||||||||
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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. |
104.
$s = 10 + {1\over 2}\sqrt 2 = \Nn{10.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group |
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. |
108.
$s = {}^{144}🔒 = \Nn{10.92591939016138}$
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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 | ||||||||||||||||||||||||||||||
|
123.
$s = {}^{8}🔒 = \Nn{11.60139984236648}$
$2449s^4-(112436+7064\sqrt{2})s^3+(1923195+221470\sqrt{2})s^2-(14552224+2321166\sqrt{2})s+41183566+8147560\sqrt{2}=0$
$2449s^8-224872s^7+8967686s^6-203140728s^5+2862024925s^4-25703580128s^3+143806409196s^2-458545364672s+638350608644=0$
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. |
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. |
128.
$s = {}^{32}🔒 = \Nn{11.82538265107608}$
$2401s^{32}-773808s^{31}+116857944s^{30}-10953322436s^{29}+710251339062s^{28}-33509506615256s^{27}+1170211058292596s^{26}-29691425943630216s^{25}+493617200842541351s^{24}-2563093770360430044s^{23}-136153511549406319200s^{22}+5167455238584576722496s^{21}-101317540269324217776128s^{20}+1192703999335242781294276s^{19}-4429951898711733045071610s^{18}-149483973523951297639523200s^{17}+4036268388799162641527101115s^{16}-59541002790231934648142695536s^{15}+630345240522567629425302746066s^{14}-5203751754819678868788726388588s^{13}+36362143340956575547968782237386s^{12}-250162817129143046903025930554608s^{11}+1972546725385414636236445891891470s^{10}-16850524678415933312186738253014836s^9+131393951082365647592910137413049289s^8-845100771668481743674765744921931432s^7+4314039623904025659790955917562980070s^6-17148739311990329953903399127228076344s^5+51998606140673855664408848072800520841s^4-116371241835830474806716427997040997408s^3+181426723023132852968436272803899334368s^2-176109616718430938071768620303187265280s+80171638196482285349976558744374366208=0$
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. |
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. |
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 | |||||||||||||||||||||||||||
|
146.
$s = {}^{8}🔒 = \Nn{12.60139984236648}$
$2449s^4-(122232+7064\sqrt{2})s^3+(2275197+242662\sqrt{2})s^2-(18745718+2785298\sqrt{2})s+57773870+10697260\sqrt{2}=0$
$2449s^8-244464s^7+10610362s^6-261806300s^5+4020285805s^4-39370448492s^3+240260012584s^2-835788126600s+1269480323300=0$
Adds an "L" to the $s(123)$ found and improved by David Ellsworth in December 2024. |
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 |
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. |
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 | ||||||||||||||||||||||||||||||
|
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. |
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. |
174.
$s = 13 + {1\over 2}\sqrt 2 = \Nn{13.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group |
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. |
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. | ||||||||||||||||||||||||||||||
|
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. |
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. |
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 |
205.
$s = {}^{32}🔒 = \Nn{14.82477941231809}$
$2401s^{32}-1004304s^{31}+196236180s^{30}-23714439252s^{29}+1973810974720s^{28}-118799295693656s^{27}+5239475702813528s^{26}-164394146951926744s^{25}+3152460689003455454s^{24}-1884976158937315692s^{23}-2345986459851718325780s^{22}+94358303978726779765624s^{21}-2128646362836938565815244s^{20}+26472288254826011788050888s^{19}+50417445455769159894431170s^{18}-10978063742803684959108505384s^{17}+290122932432608300467064251159s^{16}-4753744212995936392438627897672s^{15}+54598998036410077751874179544836s^{14}-436714300711809377426805814415712s^{13}+2266568996313109881286948921725668s^{12}-9443930478172092067066120085919936s^{11}+153569285924138821668183737305750246s^{10}-3411260012455643282533016078093314852s^9+48716027835563155199316576521575839846s^8-480634574592358844282212600776916840152s^7+3479558215819949081331510430953273130980s^6-18911274556079579556635482161438674133300s^5+77001514472742842264328954587724200333125s^4-229319366384836787105473363766786317325000s^3+473731806994674488737711970411917315943750s^2-608463749385462932355294697864370957750000s+366697674954705698636823123191988152890625=0$
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. |
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. |
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. | |||||||||||||||||||||||||||
|
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.
$s = {}^{12}🔒 = \Nn{15.60902282132495}$
$24s^6-(3008+632\sqrt{2})s^5+(143990+46836\sqrt{2})s^4-(3498934+1393560\sqrt{2})s^3+(46342192+20798868\sqrt{2})s^2-(320302448+155625644\sqrt{2})s+907787225+466763736\sqrt{2}=0$
$576s^{12}-144384s^{11}+15160736s^{10}-915791264s^9+36096990788s^8-988137514952s^7+19384693200660s^6-275725098032832s^5+2830331680270636s^4-20490408263563724s^3+99460046477154240s^2-290971312817869664s+388340875383845233=0$
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. |
231.
$s = 15 + {1\over 2}\sqrt 2 = \Nn{15.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group |
233.
$s = 8 + {11\over 2}\sqrt 2 = \Nn{15.77817459305202}$ Continues a pattern found by Frits Göbel in early 1979. Explore group |
234.
$s = {}^{32}🔒 = \Nn{15.82288272542124}$
$2401s^{32}-1081136s^{31}+227575796s^{30}-29658121472s^{29}+2666248732772s^{28}-173774204141352s^{27}+8338731572952864s^{26}-287804287422674920s^{25}+6315119157104953822s^{24}-28041637023421982020s^{23}-4200365752197925106408s^{22}+198645587687666998327868s^{21}-5061129096506078245699908s^{20}+73726267308823082442945172s^{19}-87947049283563327476026870s^{18}-27216842619744961591124737032s^{17}+835654086887645374368674033753s^{16}-15205890184479011864998900236148s^{15}+191651587009563650893651694385492s^{14}-1660301063115615092298384133605124s^{13}+8686223135028209936867163410143500s^{12}-19641905504914869202249570317433868s^{11}+287644910082291844950022660238680758s^{10}-11246330960198056495231711623386810364s^9+202355897018377179156570542109032137704s^8-2302635316094370535736927517505458541780s^7+18637198014925689744736021134333289992832s^6-111686247212255633894905670328457247786180s^5+497848050391431476660300176121042489717121s^4-1616634353567457905781858976959940153654232s^3+3632890484832687237588233953926466020404880s^2-5068769120809153666731317435277329464903680s+3315870865953295517111766154888558886723584=0$
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. |
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. |
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. | ||||||||||||||||||||||||||
|
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 $s(37)$ improvement found by David W. Cantrell in September 2002. |
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. |
262.
$s = 16 + {1\over 2}\sqrt 2 = \Nn{16.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group |
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. |
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. |
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)
$s = 14 + {5\over 2}\sqrt 2 = \Nn{17.53553390593273}$ First $s(n^2+1)$ whose best known side length is the same as $s(n^2+2)$. To construct, either add an "L" to $s(257)$ or remove a square from $s(291)$. |
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. |
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. |
296.
$s = 17 + {1\over 2}\sqrt 2 = \Nn{17.70710678118654}$ Extends the $s(52)$ found by Frits Göbel in early 1979. Explore group |
300.
$s = {}^{32}🔒 = \Nn{17.82442450194827}$
$2401s^{32}-1234800s^{31}+296028208s^{30}-43787349172s^{29}+4447593803222s^{28}-325302474537864s^{27}+17311567981201236s^{26}-645182372198475544s^{25}+13837196473272880383s^{24}+80900249230246410660s^{23}-20568679509060130505064s^{22}+920036760088872328695408s^{21}-23584191790096989323550520s^{20}+301225363617716483395354964s^{19}+3306782666397187070274017342s^{18}-284390314409319888849379255928s^{17}+8149211786420656055342585494971s^{16}-148913864740995395212738327005616s^{15}+1814416877369227440283632143889698s^{14}-12175229320483112740559170564072244s^{13}-24399018072141993476725426289195974s^{12}+1429143592810572657904567529536292992s^{11}-7892142464877656555451102305171862930s^{10}-209223038143589793338954466697147117932s^9+5720426881261352471833294168046744250065s^8-78159708093687389330697852518534676168792s^7+729681673955892025056174878918410390998726s^6-4990349834530796079653914985571847293526368s^5+25309074235966113070386092161305251964840825s^4-93449971845877531351148978361989669182305000s^3+238863061480171470191587079806750175711250000s^2-379348085850264572736715719653525693400000000s+282723151965698669809699808373173551875000000=0$
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. |
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. |
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. |
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. |
For more details, see Erich Friedman's paper on the subject: Packing Unit Squares in Squares: A Survey and New Results (or David Ellsworth's edit).
