ShortPrimalityProofs

For the purpose of this repository, a short ECPP is a sequence of integers $(p_0,A_0,x_0,m_0p_1,A_1,x_1,\ldots,m_{k-1}p_k,A_k,x_k,m_kp_{k+1})$ in which

• we have $k \ge 0$, $p_0 \ge 5$, and put $n:=\lceil \log_2 p_0\rceil$,
• the $p_i$ are odd integers and $p_1,\ldots,p_{k+1}$ have no prime factors $\le B$, where $B:=\lceil n^2/\log_2 n\rceil$, satisfying $B^2 < p_{i+1} < \sqrt{p_i}$ for $0\le i < k$ and $p_{k+1} < B^2$ (an integer below $B^2$ with no prime factor $\le B$ is either $1$ or prime),
• the $m_i$ are $B$-smooth integers with $\log_2 \mathrm{rad}(m_i) \le \lceil n/\log_2 n\rceil$ (where $\mathrm{rad}(m)$ denotes the product of the distinct prime divisors of $m$), satisfying $L_i < m_ip_{i+1} < r_iL_i$, where $L_i = q_i+1+\lfloor 2\sqrt{q_i}\rfloor$ with $q_i=\lfloor\sqrt{p_i}\rfloor$, and $r_i$ is the least prime divisor of $m_i$,
• each $A_i$ is a nonnegative integer less than $p_i$ with $\gcd(A_i^2-4,p_i)=1$,
• each $x_i$ is a nonnegative integer less than $p_i$,

such that for each $0\le i \le k$ there exist integers $B_i,y_i\in [0,p_i-1]$ with $\gcd(B_i,p_i)=1$ for which $(x_i,y_i)$ is a point of order $m_ip_{i+1}$ on the Montgomery curve $B_iy^2 = x^3 + A_ix^2 +x$ modulo every prime divisor of $p_i$.

For integers $p_0 > 3$, a short ECPP $(p_0,...)$ exists if and only if $p_0$ is prime. The size of a short ECPP is linear in $n=\log p_0$ and can be verified in quasi-quadratic time (in contrast to the quadratic size and quasi-cubic verification time for a conventional ECPP).

Format revision (August 2026). Relative to the original definition, the smoothness bound was reduced from $n^2$ to $B=\lceil n^2/\log_2 n\rceil$, a radical cap was imposed on the $m_i$, and the recursion floor was raised to $B^2$, with the terminal $p_{k+1} < B^2$ now permitted to be prime: having no prime factor $\le B$, it certifies itself by size. These changes are for the verifier's benefit: the primorial of $B$ can be built per certificate within the verification budget, its gcd with $o_i$ reveals $\mathrm{rad}(m_i)$, the radical cap bounds the order-exactness checks, and the terminal prime needs neither factoring nor a primality test -- so verification costs $O(n^2\log n)$ bit operations worst case, self-contained, with no precomputed tables. Existence for every prime $p_0\ge 5$ is unaffected ($2$-power $m_i$, $p_{k+1}=1$). All certificates below have been migrated to the revised format.

This repository contains the following resources:

• vshortECPP.py is a Python program that verifies a short ECPP in quasi-quadratic time.
• short8all.txt contains the 55,056 short ECPPs with $p_0\le 2^8$.
• short.gp is a GP script that uses SEA on random curves to search for short ECPPs (a deliberately simple demonstration implementation: clarity over speed).
• certs.csv is a list of short ECPPs for the primes listed in the table below.

For example, to search for a certificate and print it, run

echo 'printshort(nextprime(10^100))' | gp -q short.gp

Challenge

Below is a list of short ECPPs for the least prime $p>10^c$ for $c=10,20,30,\ldots,310$. Can you extend this list?

$p=10^{10}+19$,  via short.gp (<1 CPU second, 1 level).

10000000019 9322349340 1921958667 116108

$p=10^{20}+39$,  via short.gp (~1 CPU seconds, 2 levels).

100000000000000000039 89951393186720294033 26135327929659638076 11876954936 609883 131044 1915

$p=10^{30}+57$,  via short.gp (~1 CPU seconds, 2 levels).

1000000000000000000000000000057 106991342299430347297585638871 188675017969977395328028624483 9586166844055967 21068377837950 13060603184829 8314847

$p=10^{40}+121$,  via short.gp (~2 CPU seconds, 2 levels).

10000000000000000000000000000000000000121 6837934585178113515919856555491926815931 1941385895719895246468999069210638479448 123753424054361008505 23916465088426823 4639350154373931 234644419

$p=10^{50}+151$,  via short.gp (~3 CPU seconds, 2 levels).

100000000000000000000000000000000000000000000000151 73553121250676669289610938017565132196113920353122 35156187966317051233812641754435998302169352619227 11740190631638861982816855 6407187416739668112924 14970170982497869326898 376113093052

$p=10^{60}+7$,  via short.gp (~3 CPU seconds, 3 levels).

1000000000000000000000000000000000000000000000000000000000007 582594560733647942864167554683101932559817757617443321218616 485107000290978194104100671498974843211121245518759155383025 1591141197950235962428006613308 7647916346979923347916109359 1553732938852102786830521800 166971983878855 29993889638 72097552284 955641

$p=10^{70}+33$,  via short.gp (~45 CPU seconds, 3 levels).

10000000000000000000000000000000000000000000000000000000000000000000033 9193638238367761016751675951961267177328031093306760076142420392370661 5154446894383162465732893340634268054473714420328923353004710402276202 177664059059812136202711786749417049 8649705917060546335363234070525383 42605680596938635666326245539183459 303885057908347530 9987604800409038 5847492647777907 1449221531

$p=10^{80}+129$,  via short.gp (~180 CPU seconds, 3 levels).

100000000000000000000000000000000000000000000000000000000000000000000000000000129 79511867411473558312500341763373077537636135906738001200253505184066981570467039 99442333257700224273137926203143924904088958339109988959851988698298781351413343 11976780269137162447518084360610902692836 22699823929919688007129566364878825762 7101779907499577969152762520665699292 547733240424210164579 2831324762361707115 250915116616157268 3962729228

$p=10^{90}+289$,  via short.gp (~350 CPU seconds, 4 levels).

1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000289 894968576468032599378378721243828768281281629751362196980139904075779963787129703765632027 972372157545683975489770775017330129967703555095816300746070298618910208458215454650799415 1346878956075816454678411121385548900558290788 19024192904329440419320262763161156260248 20521696958027605314557996718183523621816 512517326462716280059 19267002490130813893 4847101199028585644 6455262666 748260592 553167877 37012

$p=10^{100}+267$,  via short.gp (~800 CPU seconds, 3 levels).

10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000267 9922776908302697864551893522633781539036740348871476332653810465626471659307742084010027213820226625 2836186773449440978191705565856816079355043173337102496065404967247933485306358946184015108636857285 159990589984239373593745787852818347617199852606512 989722211747899715823185566559643143324275776131 1266155285499745286234110363167600406791854970931 1875462898886971439873117 2031896458388150064946 1190299667308778102454 57552863118

$p=10^{110}+7$,  via short.gp (~6300 CPU seconds, 4 levels).

100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000007 6463512687231274453314405725127389227686510737885438111434665718277396637427547966320233024608351230230861450 94164843423036637004338287352695725396243376738721874335250825798680411803544771226508486391278261756148716303 11887982306821438660772844495374643156319942615511664684 624353010795031767996296430190839699282762020204925 455383511998255264520601283777267984573932631182927 33666714578268032448009065 1614534670079734726846 737441289820696125505 83094014116

$p=10^{120}+79$,  via short.gp (~1900 CPU seconds, 3 levels).

1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000079 132063756553348037607726246459351520545782379718387545073787885595088443428910224053262081547377506070464921665690244443 766759587882597019696391814500801893207393762520787020231259750746628420452950574472606593873044215447309872888211933399 2032407269668174614737384431875199029029184284197878747552629 1199197362586136210273591905540461080971671185565224891285 1235746649075380226211220902889073034006519549905970133438 103755726490907726500461806304 3092901622776628753789127 800991492200408803139269 9138053287529

$p=10^{130}+1113$,  via short.gp (~1800 CPU seconds, 3 levels).

10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001113 8838054562218699739469769827167347178366084538631527476350638177688407070489981665391546772154550105442112460156690474042240671153 9070442852984080644726137002845969709600170053521757279077458871926798567371260193902248343382297986025014308252012522028148277940 143023612271167604986843854406903897909541795703955557853382262527 4279311677016680608246200050783064414358575781642362807565916 2514922654280625968927317951358588049284674438860504597506761 4852063280663853797388579475379 997818570353169127828954 6427085119542794459835918 3234875012178

$p=10^{140}+13$,  via short.gp (~13700 CPU seconds, 3 levels).

100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000013 58641989056675720896672212411665741567276120290417855224286073593918069783779639337562924504400725658476104789609335447371195336334882769838 49477426077173743212373028968808725584389507235326871104842053964179679773766627681933870798077570405431407409265070868732921824050415203356 13102807158286493606695640041882330489449163784474568128908900189019724 9194143353218612803141831381890939699022032316343544916837584131946 9446062608280649498405523563298491468876161342077046283467239660178 3772692773649837804777542519049033 46939157959996065750332458522 300844653378877496051605268651 1209366579531502

$p=10^{150}+67$,  via short.gp (~11400 CPU seconds, 3 levels).

1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000067 318242351116513938119517967616301475671486714041557198261085687482277145984397246332451109797710555221454953956380848003465751737440305039178134859211 720477903566384043117699259849645626560674563858596916881593754258176690470197473175283690734334846930259881683458635858051341569534187208064885697331 3274688382283250370121362465461297838165277819878561512256860826179463643421 419710093855186023983988396848350352203085299959848344457841346392366791 2738868097477462328198525091918736337577217055081977682817874517862422856 7790490239009640360339423052989945581 165044629407820523237731842524831306 65747092211780530324214107412979080 536903698434235914 19738996086605 20356708900543 4294067863

$p=10^{160}+303$,  AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~3500 CPU seconds (6 minutes wall time on 10 cores); 4 levels).

10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000303 1459610408796319836784355481875424878795873921791031879135324980121367679324525174627177955281717548199131578517243540832793275941249177793495795195459013374976 4758106707451115267653405186080802099032088051348450733120020318362579889791887278021953210724205680736180811567428047248223479702468810707973314657809819169814 1019270628659222703537437424625804232803263645948203567236028599943201735592927793 448568730017408308873597708912961564403152659118218528354017946397296103365610 43320014997290486975014588988392937147131394524080329101272755177864009581337 700117564993251844165020303999846974095 718795565085543471731276397799125154 10968362732951735055925268568007215413 6744096990366654554 2535320783744295006 618160122705634570 4589520461

$p=10^{170}+63$,  AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~25 core-hours (4.2 hours wall time on 6 cores); 4 levels).

100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000063 93470473516953715264486130229526450936336663069233212789920054348703146246081993413887126031954487883550229314145173583538960879771857278151959126040576198335611057228717 18191774762979389705788067608435982095464290750349993788592367370329962726506894428797813319821022218482986963172035862563078186482534693386993993170270036415782051687010 18355496581896752589164838500283823405264729420962202164067182929913797513314592876438 1412501385784965102422200446057159010676876502635033859914012451612274469545278147 1026575714058769518495199429383645745069304030221667270898318990223354868198806083 71566046837985812611417285149617352963848 124391381655647142888550432419387618228 133936886864869119661578271443235443638 62042239500136966685 4350455900978038280 5154736213039779296 3555579293

$p=10^{180}+313$,  AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~5200 CPU seconds (9 minutes wall time on 10 cores); 4 levels).

1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000313 676314487666624426613264577413941673599493662940655898562057070611245460551288155535091786832776359738580085216599663216771952967980765637893778558559819437935500208875865390249244 437588306890387722589892662578409413226705638943666450657740601038982508421822919939552462866951504536517925265322210790219141179699027744923317252688881621374183616123925848009969 1166336987321735003266566202037904971880547636635894093000838667118924529020328958360209285 181230328020199888788167879924035658953885968492820660612531541546092140880882001291413109 180740652885697327946185328655571807826790152811195448489571030474428954342453352279735131 594606096126489693565799487615702466460829756 347840253512180047137361416902967552057 985111460480449331112129520819167682825 87106255631465576389 282050657021841 118686513408715 58134903

$p=10^{190}+253$,  AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~15300 CPU seconds (26 minutes wall time on 10 cores); 4 levels).

10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000253 9032059704630233730649942650536047579693026386223857885968948358385423848143230060965273666340984028744349337999958813471343224132869169824548586931328844850141135248586236633667573758081519 8672697054356251821949201159081436227181229023012836043139512670476451145681061017199275263310290576408282862769512107642454036464908582668082645931949598601721146074781572407560952323413337 119560819481213251567972071174401671086589625916629105075129472983232261018104163860364995031733 172530222172383446552035525102294162983572973133510511707335779304850506203331092191105 247541746847101909978508275336064523070746013067608908203069690583621159576001738777093 23603617419997307198195223387522299521813891 28156617424911230951215474568436310983 3407427835671628837423191861147980826 7206907075002775160 8563515044560 11196830017554 4888172

$p=10^{200}+357$,  AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~38 core-hours (3.8 hours wall time on 10 cores); 4 levels).

100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000357 57834291042150292495995563609652096282941917786582520751588842815371340027216658579432999024002709915305656128581620025390448845210225124177561529171174256999204857514501328789133046559706504654019024 11014890427879667372499868798052112552439497346445545565585187250033094195651188657919359377356221017943370375110895658588552650470341345570751912995867508645166912655969863486486372574381190051284073 450153423952056675364935857477723043101127369120523333039334166374620323711184334771772552020494112201 403197499065195869777358579896162701416242962142916541281892395320601106094138547554226079864635718 2289902501950831138158126198253297541276141821724838373701307085709523139401377355655500975010297949 87313221154928125278237123659305289785138559754195 669697371797517697784437372418971920470508070459 338761438897941287847471585101516158723344790380 2887645374971118626289461 4732035594544 19377378581309 9920677

$p=10^{210}+1129$,   AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~130 core-hours (13 hours wall time on 10 cores); 4 levels).

1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001129 557725458670982857152701997737148120074179858676347352439240212947335296245269350037271231339174404597658820084347360800831005087442219572475919810042848884392240616122455825965402365924445205637652891620237735 895936157069497478012969828777049670214767805330637988492130790733285320535472863000401234485506654546454784897774754197969294565733655057505481214814004515398195979802948975862036034983282549805209518398157914 5203093073981852709587576242855715876079437472094624099342035538570950407004773976672512290723118961283679 6095979198458864115657101500654295273083970773020165332263912745341081179584651900846763624645866775595 4643574400169070088062097708216184689442731165151169481759449284636259727879323824130267064775719725269 8888911269763810304813009748794112193722575700035107 133738894229803375450135094445249110293620632851 350913790142154716182963401103259823262540823894 24469337255187634107467641 247642374853433274467896 101528819013334583333768 806854553927

$p=10^{220}+427$,   AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~21 core-hours (1.3 hours wall time on 16 cores); 5 levels).

10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000427 3416499526622594587844071745456294587059411345982376947749872864862655405811356873224053390219028684688596817941844110868576141553673252281779845164509868398443470209474597699493403659230492827759837127387430239682944613 6006276544225210184196797915046225210185454988532059252746609191615352973711100219211316870170917525537964739128977481837395200167797768802350323065997495283235734578472382581558742939648886712147439162550237040606073401 448156923597863105134275783348497405831387676041145813066098401607950967625555918808492226059148484396393696781 356860565316986131326573141638700097057790884140550598368460484063045388342863104951589903313814876901373119 235475315602681182045794330549744758582751870575143848154563800260190045639608398331897228220162384075380062 876702734371026964729322501543915199510691082597897448 3219039467928348897979246467039628467704633714184144 630381185981968096813155230284520721691869529589426 191859686929844477754508340 823422514456264920720077 1019737062093534972518613 15708753327017 59878403381 173798812063 522901

$p=10^{230}+753$,   AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~59 core-hours (3.7 hours wall time on 16 cores); 4 levels).

100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000753 43534437444430401745762142556742195476342815888370757926592028562161261768750186054034438134682641844950329005827142216885421691640835607968023967274172939071749520009285853766046714573887761888395141794229959513556148127874160012 81554256547629276798998368178777172366730706993272025122115006538898971860179960890190450149340259212696987350330861391298192225969128417791852080885822175650024104585025668242413073470722877110621501038274860142774009007614606792 118712604394077963669475587423715104933068211984512802060778101004665250180340380644737728129406699908094997253848889 1015077950073060581199232549105888650054334797653505452586900205603597256675012880648047524989960752346056159097728 1972479326927197014447694432815254992003925708477010147346663478720079805944593459537912298701864798727353443234179 1722326225606603131591653201802152706111485816557968050612 107319031146864760555245507365800410401651699810109253953 113804162611347207770128774219414139328181708083533877111 53620144903000427810421300067 155115280914589317306361 277788472012644935344922 961294610059

$p=10^{240}+1723$,   AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~310 core-hours (1.6 hours wall time on 192 cores); 4 levels).

1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001723 247199519734287037580699410593482862389065571020433543672856683348557310444195774882596257853216355009480978672757062094817818026590735250916642101045466643319992673675921436399623686647701292405460826439925770399542902626711344475366232771 91661449377475582923215674898448627642518552812614104741899281075250543813753328355137431973172920517747428497682623256429826597808482621552880118214532717812845664087352265712025257721671982997749428873798729176008813688034534193067004357 1293211631606115275553576292879235420776378897751411190169604986425977631145229571833932955952604082440866072316495644278 92983113241736143878165912696021374472095389509412219767922117511592279865593952749194091601445094444347878205112512 545032289092865101387095047622158097548838895936093210584332361833874253536360527549175642499219638862367978733701028 8397744229309202175153202875871358166943004370193963257519941 121034850878479620279125503869396653477693425048606842637 1177137713683157137164147142831714621360837717323235552347 58474138145734262335784395546 5858530310572491129861055 9110075576390471961227929 3653788454685

$p=10^{250}+1227$,   AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~290 core-hours (1.5 hours wall time on 192 cores); 4 levels).

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$p=10^{260}+49$,   AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~170 core-hours (0.9 hours wall time on 192 cores); 4 levels).

100000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000049 83468605194508026993092816470412226520305400744910723492269887217297369632252117052050195470133238776724221262391180683867108633708280518368167104547648880945560147601214491933691147746507422236807309259199156627208603409561480101802078151017194932007907619232 32725151960299813806913869761172507513623318133846079364825922612800428167798869633683239284218095855851505662954353470096543931107728952904281570040379838560340603414014462638635343759619111883846994367622211673996878281266728677563383129094515307651666246334 1059817572042360692410356590916124966811393512187198024873160249476048366159921423390414707842843803710742859437761195264358992868153 46541905731263149506990647848184148750801436922998286941745862645314931276135743473129164392092188492557309988546971200062280541 13901137196376259685675154055973037060773837701009012548720202283377542666808621893888151415935045258983852066173916539518350988 12520006665756740388382391781761673333504498482567877034232194318 3131475757817284557529167027117526558741427302173187809337054237 3761802518224532490811107837907070857534448124479112826790455201 113791596935657179894199652797901 1014326647104665159541524024 1406579721631008122710063487 41471310100650

$p=10^{270}+361$,   AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~520 core-hours (2.7 hours wall time on 192 cores); 4 levels).

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$p=10^{280}+13$,   AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~500 core-hours (2.6 hours wall time on 192 cores); 4 levels).

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$p=10^{290}+279$,   AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~380 core-hours (2.0 hours wall time on 192 cores); 4 levels).

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$p=10^{300}+331$,   AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~380 core-hours (2.0 hours wall time on 192 cores); 5 levels).

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$p=10^{310}+733$,   AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~960 core-hours (5.0 hours wall time on 192 cores); 5 levels).

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$p=10^{320}+699$,  AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~390 core-hours (2.4 hours wall time on 192 cores); 5 levels).

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$p=10^{330}+583$,  AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~1080 core-hours (7.0 hours wall time on 192 cores); 5 levels).

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$p=10^{340}+1153$,  AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~420 core-hours (2.7 hours wall time on 192 cores); 5 levels).

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$p=10^{350}+133$,  AVS and Claude Code (Fable 5) via OneShotFastECPP/shortECPP.py (~900 core-hours (5.6 hours wall time on 192 cores); 5 levels).

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Contributors (both human and AI) are welcome to submit pull requests to this repo, provided they follow the guidelines below:

• new entries should be the least prime greater than a power of 10 larger than any currently listed;
• include the name of a human and a link to their web page (if available);
• specify the model (and effort level) of any LLM used;
• give a rough estimate of the computational resources used (e.g. CPU/GPUm/hours);
• provide a link to a GitHub repo with code that can be used to reproduce the example.