factored 135-digit Mersenne residual

Discussion

GGNFS and Msieve logs

LF(2^2281-1)-1 = 2 * 2 * 2 * 2 * 2 * 3 * 5 * 19 * 23 * 41 * 57872616793 * 271819582242082307 * 3297516038750209158351848769787405523717142645335063417537342353153 * 114432032892115797882627811961831608965139527430443490158471019890329

where LF(p) = the largest prime factor of p-1

Hundred-digit RSA moduli

Here are 100 100-digit RSA numbers: composites which factor into two equal-sized primes p and q. These numbers have the additional property that p+1, p-1, q+1, and q-1 also factor into two equal-sized primes (about 24 digits) and a few small factors (of course, including 2). Supposedly, these numbers are less vulnerable to special-number form factoring techniques.

getprime(q1,q2)=while(1,gg=q1+random(q2-q1);if(isprime(gg),break));gg
getrsa(q1,q2)=while(1,a=getprime(q1,q2);b=getprime(q1,q2);c=2*a*b+1;if(isprime(c),break));c
sz(p)=k=factorint(p+1);t=matsize(k);k[t[1]-1,1]
f1=3976353643835253325869338
f2=7071067811865475244008443
while(1,n=getrsa(f1,f2);print(n," ",sz(n)))

(Also the version which substitutes -1 for +1 in the appropriate places)

About 24 hours of search found 36000 candidates, of which the top 200 were selected and multiplied in pairs.

Of course, you should never use 100-digit RSA moduli; such numbers are almost trivial to factor these days.

1606428709899644729874675979292345859654261912647353093530637240089303053475053483933871758658410219
1704014108375258232737556170089148629948631896518967829879823541975881140640737874146502172074720287
1781303871889377223414878066041046394989792128486958233489182544753788481820611087134561070907497173
1862237772373996905689295251458388912728129050046313202782606882823454460745331091160983437356835271
1863317161108440491784402169090723710328502758082436318999831475935478843114144639261236051995685229
1928186552880330378657548107358842775976081107486737809658287620191327077197836105307352011237748301
1971657626674264204865458242316555816541138382753209294935787227304564152259974057831527487103166407
2028049839351881459899909862326608427784064033926606997582218719057211695841608494095970669094226173
2052129505036174353773730742162145457926784364221422012948102112673397423603447131330887211534074779
2089380960944832852716732137816387179950714557613898107752545827218102104231909068695806721006463341
2238146047776751329237282406300280723312108210627221616403472491269905403902480619844803342962175879
2252538625538704083309972363187504222543758486053793308473796507040808159187723618530016903811982993
2294314770434088879896648129879522862399735253256066308189801083115545474012861193891721089672893239
2323270575724253209186988716872865510807032805830056848715924486914924134600890115589480130665334411
2340254553108776924011125627381261339445389397114169080478284013493485210904286804316087215113459061
2346001386853602671211488705804568717925421375853437909727471388811164647389494301887008402196318631
2367544112365820345900961281062642258234942394443158330248804849176766550140198720330668796048781193
2420752706808924291072117874364349749755162091948721183545654812632490986483168192081530407225663921
2468766347735017772498998955290809747612377851380639323684169970301869749564513583330333464156422029
2502723145108906755656479273164618329956090686604673606063520571902575267773401392164793460633560531
2505466024287737444575216343678719806113221284101210556211381272309131582105038072202248755280181053
2514819408020339284762792563494866333400359198337490607288714701219066433909273008391279171408813909
2527070226401987187553918134898646038208041262679245433517287265153695604282864930761615170318299969
2540006700253043438302266678754859367981950533831737339343095788860764404208272622207473848591257853
2559602838320872424063839428049351141187394194969844966839972904073119853633551041094042192844596623
2623226576138861804707078029794032379793347795252756894680066223422728200102481679595705532226921801
2656546631363865286061546433080659939051141076140368745110956525896356982212769452133426782895706041
2672856585396187104631717381973028466310217523948072541704370135521290338811009221873650428067475137
2688318140234922481333122344836356763753220764122293336352722027470844031628052090912309291336938943
2759342509671618107340999944607264715474935857124362293792122016288931611236406102105747366190861569
2781769177702512671801569869962989208899399438756551577799592732169792217879872698916047917931681023
2815581847128494489899327887766685641376214823135649001116319904019384215684581183417091443118320121
2832427209912011643998408435789427498041586383561657807137281108681095870314371454323666746207185907
2845407732379697878681523334158764800840617750182117533868015227154956738880828447097298720234466393
2933578246279942472905959058552670698685457299366671355661766502296264039035282957746781846260344343
3020324503955108891023734327960331076750789722432890614898138580906827276655007213810650458844980997
3022437167285430526788613748366688899847026835117048228709385357222563460560194574651479261561373969
3044221224943319376085788968740676321823230069427609973761897582455882269139200231188981745661099039
3067356733432768441722833695795890642250999999212551526314317461672510673400216070069543355000378321
3104892984247299440400490905970668142655174877497724946313845872959040684940174832205731874619109771
3127153750038505863221407005484368161932229229374971829124297672751269265278543058243102996233395521
3138157352441741746915878674601574236118154960689259153929497969198155960143138779290480526648147337
3172669852191950198900850940738879551536991562963084637192364041513427165679798094548084131953079433
3222605460279557745165261878407998313390213960462554542476082123783721999392039488989859238622835643
3249180651995477004052348718696285774308609755015381897565655454936092764455218570778263725807266529
3264071864402627680644279058535539778896128551805042430623967642951237093724433977084393916722337731
3281923057304381024478654741215917726775160906509928984127132872958049218389565364951783098971515041
3316347985545278949148842962909688054700468084441281317962118498475265773076100016316164207339462287
3377801256714914369037381417575555077754785937493204959059364587957290949188609875675624560657411429
3377804852584007053329606486965993139285758158046823789252587442043267826059569337541596906160764141
3382614110520418856649486657541298114497291413771436042569010213428654191029959383004428589173711761
3425566082292472765785966410970564242016526964524142627572601600608895349156487611582646858700322781
3432593773258855780026935367837828098093522030747847114481033192026782716423803708875854105419552851
3449834606703586935768163097997460660810660672500108382660718955631563381526065256790265360908457323
3555084821948618983805640142935224233837633142222920194445237243791835243026051037013121242921587333
3574646437350503859361213693195328430462082341283354865866581094384690686095699543644433003913076571
3581108680106505887376502020883713477945953510763402345263494160148629534474631068842936794746748059
3632193157339998240932055524340818664435201283668694133980824106168616294089853645566965892016190707
3660984718942436040061828717597535767299661792298244678108128507341798808086740040342617323420625193
3694730049440206433517860273503267990779539690648197325084376578237026864361845894530652965634182719
3698748863697349364220828050863699461647006020001946237521591439590073305661357719421521605334278367
3740342228497990126337036972150539236712775408339845080675758157462578195023071803412406224147041797
3760386712873008018055233809234881765214954524005722082385233612101334861580512896575422129976604999
3767677096130786941657778333261372076113562370557367620595346554915074298921006318460649684815267099
3795023711379423299783064857375120869896072374252578421709750333830505806327716109059214567825133431
3799313939844983981000281632123870193575121072007543595251174467838773372094417526472590203209738621
3818194893276794667794753004689695267576378659090289470945470869585642313481157785048295037254029431
3874810275455576530281824470154469628261424410612406958135430049882587050731553541437487028612357701
3900454505476552361238424165543354565914202955353943948051418443519449100251895832664972028769154629
3905341385660724823193417033287651315122167893382830678649475883221742265778380762932205104394299333
3943111087943883109511339404868985408811254225397641961685465096178749640748411249820539444176754873
3944610641742138358460460575593233270514173116700633999964348827489387944593615430941947382538500489
3948650576194654668403044630495960385902128564702751223852502018656848090218777280005390176186830781
4029939222885065928076025523884984978382193895925989062184415443436671797365092254657645199154680737
4071167656639439354509739437261277274036493179680366718803335652522373089871838354459162560297616439
4126032416469826340211683734291505070779216465336697318408551471805921064468976443370643281189788201
4192876971876078441668792599802789644383032501530745059626629341152859068528182850991717666321686121
4195079625975933145709050784678711594871210778262909592906983169942438864975719807979098081788146409
4204602693001598658906245028238540549036761438088876115846366877890116466948492839962121083712912979
4372333094635444411404161699136339965599973060347379346779270139251189734297128428275138570975754219
4390240481760304098497816473597383527293732492932531090531192893488177102504849562909714619103642921
4411885605137661379932684231165551148457776975073400165085423075316520416744207983984125827711433699
4536255981065388269463781807246369361368486703084950967874979915733032966466736438902723804366067109
4616001349352079571132689963661844312241282729063121403411497199021977141930876336430831724471619943
4719681597334134304867170103057066130307898945139038600636339045903622581613921660556363691040022803
4746063361550105570699647738423983407551144991697248833824237588343392890308963978423101130872876551
4826948064208647142259931668996899330815850154136782810008953449218549983348314039069784382413548857
4831870377807201563223533271698875622456079721031312891278799444113782680890073855645179449956046899
4835937311773458816746523449714776859486220074958122150329976048049943095503000785610919784084612009
4998911338819591528184578091279141938809778087943457933605305150671037946483775726827743063554020327
5209088823969632804859212804747861066535659452150144864490963356438368941799349329545566810516011157
5478114759049141052504652826554630032413669904251036294616158721816323754865718160763366847607600809
5816255086039065593822318338282112418463727602219891827563207282582426320308504390732473216253915167
5846112823892760213723927356664281463847091564214073509950235987714188563476697540009734390145419793
5853215074287720382025823189503080595986472048459799918166737862968671105664724455619189156382073467
6029669892554445009749222158834204895410314886381788933780537144624468333428607399120604344928354609
6190084975945507543072414355969653287399527511265226494946581136594995886693792721186444801198819551
6403544569031008763054654773021400705543445146362286250629708514339353817758169087777504652234269589
7507602920570761337713972258731432878488987240251459558778594444861549373115673133962887949623141253
8418059742469774751246918389821786647059804771789656108322744613592280124246009891390922294272758861

RSA distributions

What distribution must I sample from so that the distribution of x*y is uniform between [a,b], where x and y are independently sampled from the distribution?

Update: I've been informed (from -c help -i probability) that the answer is 1/sqrt(-pi log x) 0<x<1.

Triggers a Euler-identity-like surprise: where did "pi" and "e" (in "log") come from?!

xkcd - A Webcomic - Substitute

xkcd - A Webcomic - Substitute #2 might be a good basis for a game.

Another Mersenne factorization

(Continuing Mersenne Factorization)

? ispseudoprime(2^4423-1)
%44 = 1

? print(factorint(2^4423-2))

[2, 1; 3, 2; 7, 1; 23, 1; 67, 2; 89, 1; 683, 1; 1609, 1; 2011, 1; 4423, 1; 9649, 1; 13267, 1; 20857, 1; 22111, 1; 39799, 1; 283009, 1; 599479, 1; 6324667, 1; 7327657, 1; 193707721, 1; 12148690313, 1; 12371522263, 1; 361859649163, 1; 761838257287, 1; 6713103182899, 1; 224134035919267, 1; 3556355492892313, 1; 5157050159173695487, 1; 17153597302151518561, 1; 17904041241938148871927, 1; 59151549118532676874448563, 1; 1647072866431538116058878617811, 1; 87449423397425857942678833145441, 1; 1963672214729590922916323781834466879, 1; 49929707724752567469731915956762751258933207272739486748238351859309991348433, 1; 40393566547943595749562506243285884534929026356774912763863482259566537671583290150415083011252727505582091, 1; 245646981125691497673324668265536334044341262452177697864695233686173498977525877540362298849614068695233671, 1; 29792282327632127192280512714312339494458105715740509816040019161219528270861465666941470299423164525021764760664757557501816665197191248140710453823079834899917278481203481942074120698987141443607970695192539694488469929529584413885826254451155851081784465332583575562462448913571987013144129130422035667076921, 1; 81306434126435390369376308017426816467338589074376606953450887738659949122190481971292960301001128628269985908910250733571484380927682097166969483636698401864705393738719321415525908871375830643489767976984133538274257006197857712319103629206245907496601803359738478994241731519997695185318284051254656008033048015655006605203258597365919579712675545019366698923697429439095730189943, 1]

? znprimroot(2^4423-1)
%45 = Mod(7, 285542542228279613901563566102164008326164238644702889199247456602284400390600653875954571505539843239754513915896150297878399377056071435169747221107988791198200988477531339214282772016059009904586686254989084815735422480409022344297588352526004383890632616124076317387416881148592486188361873904175783145696016919574390765598280188599035578448591077683677175520434074287726578006266759615970759521327828555662781678385691581844436444812511562428136742490459363212810180276096088111401003377570363545725120924073646921576797146199387619296560302680261790118132925012323046444438622308877924609373773012481681672424493674474488537770155783006880852648161513067144814790288366664062257274665275787127374649231096375001170901890786263324619578795731425693805073056119677580338084333381987500902968831935913095269821311141322393356490178488728982288156282600813831296143663845945431144043753821542871277745606447858564159213328443580206422714694913091762716447041689678070096773590429808909616750452927258000843500344831628297089902728649981994387647234574276263729694848304750917174186181130688518792748622612293341368928056634384466646326572476167275660839105650528975713899320211121495795311427946254553305387067821067601768750977866100460014602138408448021225053689054793742003095722096732954750721718115531871310231057902608580607)

This is the result of the factorization of 2^2211+1 noted earlier, and a fortuitously easy factorization of the primitive part of 2^2211-1 (and of course lots of hard work by others for 2^737+-1)

factored 2^2211+1

? print(factorint(2^2211+1))

[3, 2; 67, 2; 683, 1; 2011, 1; 9649, 1; 13267, 1; 20857, 1; 283009, 1; 6324667, 1; 7327657, 1; 361859649163, 1; 6713103182899, 1; 224134035919267, 1; 3556355492892313, 1; 17153597302151518561, 1; 59151549118532676874448563, 1; 1647072866431538116058878617811, 1; 49929707724752567469731915956762751258933207272739486748238351859309991348433, 1; 40393566547943595749562506243285884534929026356774912763863482259566537671583290150415083011252727505582091, 1; 29792282327632127192280512714312339494458105715740509816040019161219528270861465666941470299423164525021764760664757557501816665197191248140710453823079834899917278481203481942074120698987141443607970695192539694488469929529584413885826254451155851081784465332583575562462448913571987013144129130422035667076921, 1]

Run 605 out of 0:

Using B1=1755043, B2=2140281790, polynomial Dickson(6), sigma=1856383114

Step 1 took 167035ms

********** Factor found in step 1: 1647072866431538116058878617811

Found probable prime factor of 31 digits: 1647072866431538116058878617811

Probable prime cofactor 29792282327632127192280512714312339494458105715740509816040019161219528270861465666941470299423164525021764760664757557501816665197191248140710453823079834899917278481203481942074120698987141443607970695192539694488469929529584413885826254451155851081784465332583575562462448913571987013144129130422035667076921 has 311 digits

Consecutive sums of squares

Inspired by Abstruse Goose, identities between consecutive sums of squares are not that uncommon. (See 2030 for another example of consecutiveness across the equals sign)

25
= 3²+ 4²
= 5²

365
= 10²+ 11²+ 12²
= 13²+ 14²

841
= 20²+ 21²
= 29²

1405
= 7²+ 8²+ 9²+ 10²+ 11²+ 12²+ 13²+ 14²+ 15²+ 16²
= 26²+ 27²

1730
= 6²+ 7²+ 8²+ 9²+ 10²+ 11²+ 12²+ 13²+ 14²+ 15²+ 16²+ 17²
= 23²+ 24²+ 25²

2030
= 21²+ 22²+ 23²+ 24²
= 25²+ 26²+ 27²

3281
= 5²+ 6²+ 7²+ 8²+ 9²+ 10²+ 11²+ 12²+ 13²+ 14²+ 15²+ 16²+ 17²+ 18²+ 19²+ 20²+ 21²
= 40²+ 41²

3655
= 8²+ 9²+ 10²+ 11²+ 12²+ 13²+ 14²+ 15²+ 16²+ 17²+ 18²+ 19²+ 20²+ 21²+ 22²
= 25²+ 26²+ 27²+ 28²+ 29²

3740
= 6²+ 7²+ 8²+ 9²+ 10²+ 11²+ 12²+ 13²+ 14²+ 15²+ 16²+ 17²+ 18²+ 19²+ 20²+ 21²+ 22²
= 18²+ 19²+ 20²+ 21²+ 22²+ 23²+ 24²+ 25²

4510
= 15²+ 16²+ 17²+ 18²+ 19²+ 20²+ 21²+ 22²+ 23²+ 24²+ 25²
= 28²+ 29²+ 30²+ 31²+ 32²

4705
= 17²+ 18²+ 19²+ 20²+ 21²+ 22²+ 23²+ 24²+ 25²+ 26²
= 48²+ 49²

4760
= 8²+ 9²+ 10²+ 11²+ 12²+ 13²+ 14²+ 15²+ 16²+ 17²+ 18²+ 19²+ 20²+ 21²+ 22²+ 23²+ 24²
= 23²+ 24²+ 25²+ 26²+ 27²+ 28²+ 29²

4900
= 1²+ 2²+ 3²+ 4²+ 5²+ 6²+ 7²+ 8²+ 9²+ 10²+ 11²+ 12²+ 13²+ 14²+ 15²+ 16²+ 17²+ 18²+ 19²+ 20²+ 21²+ 22²+ 23²+ 24²
= 70²

4900 is special. 1..24=70.

Here are some triplets.

20449
= 7²+ 8²+ 9²+ 10²+ 11²+ 12²+ 13²+ 14²+ 15²+ 16²+ 17²+ 18²+ 19²+ 20²+ 21²+ 22²+ 23²+ 24²+ 25²+ 26²+ 27²+ 28²+ 29²+ 30²+ 31²+ 32²+ 33²+ 34²+ 35²+ 36²+ 37²+ 38²+ 39²
= 38²+ 39²+ 40²+ 41²+ 42²+ 43²+ 44²+ 45²+ 46²+ 47²+ 48²
= 143²

147441
= 18²+ 19²+ 20²+ 21²+ 22²+ 23²+ 24²+ 25²+ 26²+ 27²+ 28²+ 29²+ 30²+ 31²+ 32²+ 33²+ 34²+ 35²+ 36²+ 37²+ 38²+ 39²+ 40²+ 41²+ 42²+ 43²+ 44²+ 45²+ 46²+ 47²+ 48²+ 49²+ 50²+ 51²+ 52²+ 53²+ 54²+ 55²+ 56²+ 57²+ 58²+ 59²+ 60²+ 61²+ 62²+ 63²+ 64²+ 65²+ 66²+ 67²+ 68²+ 69²+ 70²+ 71²+ 72²+ 73²+ 74²+ 75²+ 76²
= 29²+ 30²+ 31²+ 32²+ 33²+ 34²+ 35²+ 36²+ 37²+ 38²+ 39²+ 40²+ 41²+ 42²+ 43²+ 44²+ 45²+ 46²+ 47²+ 48²+ 49²+ 50²+ 51²+ 52²+ 53²+ 54²+ 55²+ 56²+ 57²+ 58²+ 59²+ 60²+ 61²+ 62²+ 63²+ 64²+ 65²+ 66²+ 67²+ 68²+ 69²+ 70²+ 71²+ 72²+ 73²+ 74²+ 75²+ 76²+ 77²
= 85²+ 86²+ 87²+ 88²+ 89²+ 90²+ 91²+ 92²+ 93²+ 94²+ 95²+ 96²+ 97²+ 98²+ 99²+ 100²+ 101²

3198550
= 1²+ 2²+ 3²+ 4²+ 5²+ 6²+ 7²+ 8²+ 9²+ 10²+ 11²+ 12²+ 13²+ 14²+ 15²+ 16²+ 17²+ 18²+ 19²+ 20²+ 21²+ 22²+ 23²+ 24²+ 25²+ 26²+ 27²+ 28²+ 29²+ 30²+ 31²+ 32²+ 33²+ 34²+ 35²+ 36²+ 37²+ 38²+ 39²+ 40²+ 41²+ 42²+ 43²+ 44²+ 45²+ 46²+ 47²+ 48²+ 49²+ 50²+ 51²+ 52²+ 53²+ 54²+ 55²+ 56²+ 57²+ 58²+ 59²+ 60²+ 61²+ 62²+ 63²+ 64²+ 65²+ 66²+ 67²+ 68²+ 69²+ 70²+ 71²+ 72²+ 73²+ 74²+ 75²+ 76²+ 77²+ 78²+ 79²+ 80²+ 81²+ 82²+ 83²+ 84²+ 85²+ 86²+ 87²+ 88²+ 89²+ 90²+ 91²+ 92²+ 93²+ 94²+ 95²+ 96²+ 97²+ 98²+ 99²+ 100²+ 101²+ 102²+ 103²+ 104²+ 105²+ 106²+ 107²+ 108²+ 109²+ 110²+ 111²+ 112²+ 113²+ 114²+ 115²+ 116²+ 117²+ 118²+ 119²+ 120²+ 121²+ 122²+ 123²+ 124²+ 125²+ 126²+ 127²+ 128²+ 129²+ 130²+ 131²+ 132²+ 133²+ 134²+ 135²+ 136²+ 137²+ 138²+ 139²+ 140²+ 141²+ 142²+ 143²+ 144²+ 145²+ 146²+ 147²+ 148²+ 149²+ 150²+ 151²+ 152²+ 153²+ 154²+ 155²+ 156²+ 157²+ 158²+ 159²+ 160²+ 161²+ 162²+ 163²+ 164²+ 165²+ 166²+ 167²+ 168²+ 169²+ 170²+ 171²+ 172²+ 173²+ 174²+ 175²+ 176²+ 177²+ 178²+ 179²+ 180²+ 181²+ 182²+ 183²+ 184²+ 185²+ 186²+ 187²+ 188²+ 189²+ 190²+ 191²+ 192²+ 193²+ 194²+ 195²+ 196²+ 197²+ 198²+ 199²+ 200²+ 201²+ 202²+ 203²+ 204²+ 205²+ 206²+ 207²+ 208²+ 209²+ 210²+ 211²+ 212²
= 127²+ 128²+ 129²+ 130²+ 131²+ 132²+ 133²+ 134²+ 135²+ 136²+ 137²+ 138²+ 139²+ 140²+ 141²+ 142²+ 143²+ 144²+ 145²+ 146²+ 147²+ 148²+ 149²+ 150²+ 151²+ 152²+ 153²+ 154²+ 155²+ 156²+ 157²+ 158²+ 159²+ 160²+ 161²+ 162²+ 163²+ 164²+ 165²+ 166²+ 167²+ 168²+ 169²+ 170²+ 171²+ 172²+ 173²+ 174²+ 175²+ 176²+ 177²+ 178²+ 179²+ 180²+ 181²+ 182²+ 183²+ 184²+ 185²+ 186²+ 187²+ 188²+ 189²+ 190²+ 191²+ 192²+ 193²+ 194²+ 195²+ 196²+ 197²+ 198²+ 199²+ 200²+ 201²+ 202²+ 203²+ 204²+ 205²+ 206²+ 207²+ 208²+ 209²+ 210²+ 211²+ 212²+ 213²+ 214²+ 215²+ 216²+ 217²+ 218²+ 219²+ 220²+ 221²+ 222²+ 223²+ 224²+ 225²+ 226²
= 225²+ 226²+ 227²+ 228²+ 229²+ 230²+ 231²+ 232²+ 233²+ 234²+ 235²+ 236²+ 237²+ 238²+ 239²+ 240²+ 241²+ 242²+ 243²+ 244²+ 245²+ 246²+ 247²+ 248²+ 249²+ 250²+ 251²+ 252²+ 253²+ 254²+ 255²+ 256²+ 257²+ 258²+ 259²+ 260²+ 261²+ 262²+ 263²+ 264²+ 265²+ 266²+ 267²+ 268²+ 269²+ 270²+ 271²+ 272²+ 273²+ 274²+ 275²

#!perl -w $max=400; for$start(1..$max){ for$end($start..$max){ $sum=0; for$i($start..$end){ $sum+=$i*$i} push @{$h{$sum}},"$start $end"; } } for(sort {$a<=>$b} keys%h){ $count=@{$h{$_}}; if ($count<=1){ next; } print "<p>$_"; for(@{$h{$_}}){ print "<br>= "; die unless ($start,$end)=/(\d+) (\d+)/; for$i($start..$end){ print "+ " unless ($i == $start); print "$i²"; } } print "</p>\n"; }

Mersenne factorization

Continuing this project with larger Mersenne primes.

The recent factorization of 2^1101+1 by Silverman (page 108, entry 5623 of the Cunningham Project) gives us

? isprime(2^2203-1)

%59 = 1

? print(factor(2^2203-2))

[2, 1; 3, 2; 7, 1; 2203, 1; 12479, 1; 19819, 1; 28627, 1; 79273, 1; 51791041, 1; 78138581882953, 1; 146264881313513, 1; 20837062885084633147, 1; 258977744356523549983, 1; 301311116540899114446723859201, 1; 883533090360873723903538281367, 1; 460233616861852066165180033789571, 1; 1636198597169607245088331633873083979, 1; 19755740081951910036006278827509875120092863638283602681, 1; 79073321945266228838262115990524608069915168947411523193620486700648001, 1; 13307293644989926855503183079276784290565688582692135767130945147226822516757076633431114901443896343698334635540215594678754538286654095364149897, 1; 711718443060888357455104383759579899185453159253854240850359788937324328008225366876777905349283339583535597500393178373807851032788989008946432082299780350922963303, 1]

After 232125395706826190501, leaves us challenges:

c105 has factor 1165410983472064229956077783033756619354709 (via ECM) and the larger one falls to 60716055995924620572719451834239 followed by 115092709908691724294367540635071

? ispseudoprime(2^3217-1)

%5 = 1

? print(factor(2^3217-2))

[2, 1; 3, 2; 5, 1; 7, 1; 13, 1; 17, 1; 97, 1; 241, 1; 257, 1; 269, 1; 673, 1; 1609, 1; 2011, 1; 3217, 1; 4289, 1; 9649, 1; 10453, 1; 22111, 1; 75041, 1; 132661, 1; 192961, 1; 6324667, 1; 7327657, 1; 15152453, 1; 42875177, 1; 58846369, 1; 193707721, 1; 209898673, 1; 214473433, 1; 2559066073, 1; 71848008781, 1; 175132692529, 1; 761838257287, 1; 6713103182899, 1; 6925799047681, 1; 9739278030221, 1; 21402380066017, 1; 333808138537249, 1; 29493338799546784993, 1; 59151549118532676874448563, 1; 36360649135813582804156044289, 1; 133304540580175280319733184641, 1; 15704900959651293774270521395753, 1; 87449423397425857942678833145441, 1; 22845623493785363787359045076091739713, 1; 311532723090035002320027134588551176448471666435989092897, 1; 1113767094422199900605896348724787045161997478687751948513969, 1; 163309771760986620014536331036091658009630202169109324871656084993557377301963746856626689, 1; 655065662598900362767578455116791254630312910908827632908253411351301525671199788835777321465298977, 1; 740887329174524327455005745661306728131406389630924979029857723424410009046767537884650900512703207784774794207122435593649368728149684897691507733781127878673273900985500310966336106589083780932968731424118506572640082702736435521, 1]

using http://www.euronet.nl/users/bota/medium-p-even4k.txt (Arjen Bot) for 2^1608+1. This page was found by google searching "58846369" which I found as a factor.

The C187 (after 142724002520786353) is

1068240625262629567200191016033534686294973766625910182558848952050153878490870601823590709054333618002131347133455436225486745521448263699420827688564971719076145154668647793187749244171

MLB Least Valuable Player

After the rest of your team worked so hard to get as far as it did, this award goes to the player who cost the team the World Series, either the World Series itself (for the World Series loser), or the previous league championship series. Capriciously, this goes to the pitcher who allowed score the difference-making run of the clinching game, unless there was some fielding error or extenuating circumstance on the play. It is theoretically possible that the award goes to a manager (intentionally walking in the difference-making run with the bases loaded). And the award goes to:

AL: Chad Bradford, Tampa Bay (gave up unearned run in World Series Game 5)

NL: Chad Billingsley, LA Dodgers (gave up the second run in the NL Championship Series Game 5)

They are both named Chad.

Next supercontinent

New Scientist, October 2007

Someone should turn this into awesome Long Bet, whether (before A.D. 300000000) North America will collide to the west with Asia or Australia (as a result of the mid-Atlantic ridge continuing to open), or whether it will collide to the east with Europe/Africa/Asia (as a result of a new Atlantic subduction zone).

Kriegspiel go and chess variants

Kriegspiel chess

The following ideas are independent:

Opponent's pieces become visible if attacking.

Opponent's pieces become visible if attacked.

Visibility takes effect not immediately, but the subsequent move.

At the beginning of the game, each player may choose any Chess960 starting position.

Kriegspiel go 囲碁

Opponent's pieces become visible if adjacent.

Opponent's connected groups become visible if adjacent.

Earth image

The area of the earth is 5.1*10^14 m^2, so uncompressed one-meter resolution imagery would be 1500 terabytes. If the earth were flat, it would be a square image 23,000,000 pixels on a side.

Flat Big Bang

It is easy to imagine the Big Bang for a universe with positive curvature (closed universe): imagine a balloon inflating and dots on its surface becoming further apart as space itself expands. Then jump to a 3-sphere.

But I cannot imagine how a Big Bang of a flat or negative-curvature universe could work. In one moment, the universe is zero-dimensional thus finite, the next moment, it is infinite by virtue of its flatness, nevertheless, the universe grows in size. How can the infinite grow?

If the universe is flat, does it have infinite mass?

Customers swapping IP addresses

Give me an ISP which provides an interface for customers to swap IP addresses with each other unbeknownst to the ISP to thwart tracking, or to choose and change their IP address in a way that the ISP does not know what address they chose.

Consumers not spending

If consumers are not spending due to worries about their future personal finances, then they must be saving, which is not necessarily bad for the economy. In fact, it might help to bring financial institutions back into the black.

If consumers are not spending because of actual loss of income, then it is all right that they are not spending credit, after all, it was excessive easy credit that got us into this mess in the first place.

Certainly it must be possible for economists to pretty easily see the amount of consumer saving, assuming we are not all keeping it under our mattresses.

Computer chess time and cores

When comparing computer chess programs against one another, one can do a few things that cannot be done comparing humans against each other. One can give the weaker program a greater and greater time advantage until parity is reached (ponder off). One can adjust the number of cores a program gets and compare its performance between a greater number of cores versus a greater amount of time, i.e., its parallelization efficiency.

These experiments are more interesting than the simple numerical rating that chess engine rating sites now employ.

Chess color advantage

Most round-robin chess tournaments have an even number of participants, so that everyone plays each round. This means that everyone plays an odd number of games. This means that players play more of one color than the other. This introduces an imbalance due to the advantage of playing white.

How often is the winner of a tournament someone who had more whites than blacks?

Capture the flag

A chess variant which where the king cannot move or capture. It is a "flag".

Another variant: the one move the king can do is castle.

ESRB Makes Games Sound Awesome

http://www.giantbomb.com/news/esrb-makes-games-sound-awesome/506/

http://www.esrb.org/

http://games.slashdot.org/games/08/11/13/012221.shtml

On one hand, game publishers have incentive to make their games sound more tame, in order to appeal to more parents. One the other hand, they have incentive to make it sound less tame, in order to appeal to adults who buy the game for themselves based on the description. Due to this odd intersection of incentives, they might actually achieve a balanced truthful description, which is rare in marketing. But they probably won't.

bash.org humor

bash.org is a wonderfully hand-classified training set for machine learning of humor.

Seven-segment display for kanji

It is a standard exercise to figure out the combinational logic that takes 4 bits of binary-coded decimal and outputs each of the segments of a numeric seven-segment display. No doubt someone has achieved the minimum number of NAND gates.

There are fun tricks one can play by increasing the levels of logic as well as sharing logic between between segments.

Consider the same problem, but on a much larger scale, given input of a CJK Unihan Unicode code point, produce a rasterized representation of the Chinese character. What is even the ballpark minimum number of NAND gates needed? It's a hard optimization problem.

Of course, the practical way to do this is with a ROM.

Mandelbrot set intersection decision problem

Is the intersection of the Mandelbrot set with a given region of the complex plane, say rectangular, non-empty? What is the computational complexity of this problem? How about instead of the Mandelbrot set exactly, we deal with the set of points which do not escape after, say, 1000 iterations?

Given the theorem that there are no isolated islands in the Mandelbrot set, it might be possible that this problem is solvable rapidly.

Land speed record

The ultimate land speed record is probably something near c, but not exactly c because that requires a perfect vacuum. What is the fastest speed ever measured of an object on Earth, measured with enough precision to distinguish it from other contenders for the record? Is it light? Or is it particles in a particle accelerator?

Ice cream sleeve

An insulated sleeve or wrapper to fit around a store-bought pint of ice cream so it does not melt while you eat it.

Don't just vote

I begin to feel that the problem with American democracy is not that there aren't enough people voting, but that there aren't enough people running for office.

Questions

If, as President, you should discover actions by our government perpretrated by a previous administration, that, if revealed, would be horribly damaging to America's reputation, will you continue to participate in the coverup?

What does freedom mean to you?

Camcorder with satellite uplink

Giving the victims of oppression, aggression, and brutality a video camera with a satellite uplink changes the adage that history is written by the victors, and therefore ultimately may change the world towards peace.

I imagine maybe thousands of such cameras being air dropped to a place like Congo. Perhaps there is genocide going on there. Instead of sending in troops to stop the genocide, we send cameras to document what is going on, changing the incentives and rewards of the genociders even if they succeed.

There are some difficult technical problems that must be solved. How low-power can one make such a camcorder? The live satellite uplink may consume a lot of power, too, though that may be compensated for by a larger antenna on the satellites and error-correcting codes. Yes, satellites plural because probably a constellation in low earth orbit will be needed because of the low power requirement of the transmitting cameras.

The cameras must transmit over an open protocol so that no one entity that controls a constellation may censor the broadcast. There may be several constellations controlled by different, perhaps competing entities. Spread spectrum must also be employed so that the enemy cannot jam the transmission. These two requirements may contradict each other.

In certain hot zones balloons or mesh networks may be effective for broadcasting and receiving the transmissions.

Cameras should have GPS in order to location stamp and time stamp the transmissions, but must have the ability to turn the GPS off. GPS also consumes a lot of power.

Upon receipt of the broadcast the video must be immediately pushed into a censorship resistant medium like Freenet.

The storage requirements of all these videos is staggering.

Balk

What would baseball be like if the balk rule were eliminated? There would be more deception and trickery by the pitcher and I think that would make the game more entertaining.