Factorizations of 99...99599...99

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Factorizations of 99...99599...99	2005-12-28(Wed) 17:39

Last update

Dec 28, 2005 17:39 JST

Sequence

5, 959, 99599, 9995999, 999959999, ...

General term

10²ⁿ⁺¹-4·10ⁿ-1

Prime numbers

10¹⁷⁷-4·10⁸⁸-1 = (9)₈₈5(9)₈₈_<177> is prime. (Patrick De Geest / Sep 23, 2002)
10²²⁵-4·10¹¹²-1 = (9)₁₁₂5(9)₁₁₂_<225> is prime. (Patrick De Geest / Sep 23, 2002)
10³⁹⁷-4·10¹⁹⁸-1 = (9)₁₉₈5(9)₁₉₈_<397> is prime. (Patrick De Geest / Sep 23, 2002)
10¹²⁴⁵-4·10⁶²²-1 = (9)₆₂₂5(9)₆₂₂_<1245> is prime. (Jeff Heleen / Oct 2, 2002)
10⁸⁴⁵⁷-4·10⁴²²⁸-1 = (9)₄₂₂₈5(9)₄₂₂₈_<8457> is prime. (Patrick De Geest / Oct 4, 2002)
10²⁰¹⁰⁵-4·10¹⁰⁰⁵²-1 = (9)₁₀₀₅₂5(9)₁₀₀₅₂_<20105> is prime. (Daniel Heuer / Mar 28, 2001)

References:

Palindromic Wing Primes (Patrick De Geest)
A114646 (On-Line Encyclopedia of Integer Sequences)

Condition

n≤75

Status

Completed up to n=50. (Aug 4, 2004)
Completed up to n=75. (Aug 13, 2005)

Factorization results

10³-4·10¹-1 =: 959; = 7 · 137

10⁵-4·10²-1 =: 99599; = 137 · 727

10⁷-4·10³-1 =: 9995999; = 13 · 768923

10⁹-4·10⁴-1 =: 999959999; = 2137 · 467927

10¹¹-4·10⁵-1 =: 99999599999_<11>; = 13 · 43 · 178890161

10¹³-4·10⁶-1 =: 9999995999999_<13>; = 31 · 113 · 2854694833_<10>

10¹⁵-4·10⁷-1 =: 999999959999999_<15>; = 7 · 47 · 2957 · 1027904483_<10>

10¹⁷-4·10⁸-1 =: 99999999599999999_<17>; = 2141 · 46707146006539_<14>

10¹⁹-4·10⁹-1 =: 9999999995999999999_<19>; = 13 · 137 · 12251 · 458315495129_<12>

10²¹-4·10¹⁰-1 =: 999999999959999999999_<21>; = 137 · 1579 · 21499 · 388111 · 554017

10²³-4·10¹¹-1 =: 99999999999599999999999_<23>; = 13 · 173 · 43787 · 892351 · 1137966923_<10>

10²⁵-4·10¹²-1 =: 9999999999995999999999999_<25>; = 499 · 1790557 · 11192092829389193_<17>

10²⁷-4·10¹³-1 =: 999999999999959999999999999_<27>; = 7 · 193² · 50909 · 75334303200659677_<17>

10²⁹-4·10¹⁴-1 =: 99999999999999599999999999999_<29>; = 458973066229_<12> · 217877708645556131_<18>

10³¹-4·10¹⁵-1 =: 9999999999999995999999999999999_<31>; = 13 · 47 · 223 · 97561231 · 752275044816094693_<18>

10³³-4·10¹⁶-1 =: 999999999999999959999999999999999_<33>; = 6113 · 525961 · 311022681819554447797543_<24>

10³⁵-4·10¹⁷-1 =: 99999999999999999599999999999999999_<35>; = 13² · 137 · 37621739 · 114803165205817843187597_<24>

10³⁷-4·10¹⁸-1 =: 9999999999999999995999999999999999999_<37>; = 137 · 497888889033997_<15> · 146604397763419254691_<21>

10³⁹-4·10¹⁹-1 =: 999999999999999999959999999999999999999_<39>; = 7 · 67 · 153146215961091013_<18> · 13922617340990150567_<20>

10⁴¹-4·10²⁰-1 =: 99999999999999999999599999999999999999999_<41>; = 211949 · 471811615058339506200076433481639451_<36>

10⁴³-4·10²¹-1 =: 9999999999999999999995999999999999999999999_<43>; = 13² · 31 · 4973347 · 6054617 · 40211671 · 1576391349924455629_<19>

10⁴⁵-4·10²²-1 =: 999999999999999999999959999999999999999999999_<45>; = 11299 · 622138507 · 142256758560010135477366785356543_<33>

10⁴⁷-4·10²³-1 =: 99999999999999999999999599999999999999999999999_<47>; = 13 · 677 · 6899 · 1296733567_<10> · 1270080011051171710148107842203_<31>

10⁴⁹-4·10²⁴-1 =: 9999999999999999999999995999999999999999999999999_<49>; = 2068133234617833654091_<22> · 4835278420467857664597144989_<28>

10⁵¹-4·10²⁵-1 =: 999999999999999999999999959999999999999999999999999_<51>; = 7 · 137 · 661 · 174169 · 44753820497_<11> · 202385297440747737276100207157_<30>

10⁵³-4·10²⁶-1 =: 99999999999999999999999999599999999999999999999999999_<53>; = 43 · 137 · 269 · 63104262756053434165531315553496954272758079081_<47>

10⁵⁵-4·10²⁷-1 =: 9999999999999999999999999995999999999999999999999999999_<55>; = 13 · 6379 · 25847307363341903_<17> · 4665398423502992195279591215935679_<34>

10⁵⁷-4·10²⁸-1 =: 999999999999999999999999999959999999999999999999999999999_<57>; = 107 · 1019 · 44017 · 1316175517_<10> · 158309796516404792290681726538046377027_<39>

10⁵⁹-4·10²⁹-1 =: 99999999999999999999999999999599999999999999999999999999999_<59>; = 13 · 883 · 8711560240439062636118128756825507448384005575398553881_<55>

10⁶¹-4·10³⁰-1 =: 9999999999999999999999999999995999999999999999999999999999999_<61>; = 70271 · 142306214512387755973303354157419134493603335657668170369_<57>

10⁶³-4·10³¹-1 =: 999999999999999999999999999999959999999999999999999999999999999_<63>; = 7 · 21013 · 62357796923_<11> · 305282330734468349_<18> · 357125998293074036157207127307_<30>

10⁶⁵-4·10³²-1 =: 99999999999999999999999999999999599999999999999999999999999999999_<65>; = 51683 · 316241 · 6118347088742694034859497781562056758199779996805773733_<55>

10⁶⁷-4·10³³-1 =: 9999999999999999999999999999999995999999999999999999999999999999999_<67>; = 13 · 137 · 2241472133_<10> · 5739525849135414571_<19> · 436442184128509132538745347944168853_<36>

10⁶⁹-4·10³⁴-1 =: 999999999999999999999999999999999959999999999999999999999999999999999_<69>; = 137 · 5659508681_<10> · 3807585650264987363579_<22> · 338727940062478941662472044023278973_<36>

10⁷¹-4·10³⁵-1 =: 99999999999999999999999999999999999599999999999999999999999999999999999_<71>; = 13 · 8237 · 1605209 · 4483537 · 50291700108583_<14> · 2580114149148249055140122245523013509561_<40>

10⁷³-4·10³⁶-1 =: 9999999999999999999999999999999999995999999999999999999999999999999999999_<73>; = 31 · 131507747019752954790816626929_<30> · 2452940244751037403306006357162582158474801_<43>

10⁷⁵-4·10³⁷-1 =: 999999999999999999999999999999999999959999999999999999999999999999999999999_<75>; = 7 · 2699 · 1085627 · 36495883 · 91121813 · 14660615771925344670269790464657113371195527531671_<50>

10⁷⁷-4·10³⁸-1 =: 99999999999999999999999999999999999999599999999999999999999999999999999999999_<77>; = 61 · 1151 · 2027 · 980491 · 716634118922624678967055583729547233783668686133478376168911837_<63>

10⁷⁹-4·10³⁹-1 =: 9999999999999999999999999999999999999995999999999999999999999999999999999999999_<79>; = 13 · 4623322337637763_<16> · 1872677320577553907_<19> · 30850310090034013207_<20> · 2879917120583141008607429_<25>

10⁸¹-4·10⁴⁰-1 =: 999999999999999999999999999999999999999959999999999999999999999999999999999999999_<81>; = 2538281234207_<13> · 14627176672631679241_<20> · 29322024270098527103_<20> · 918556352796788574399718334759_<30>

10⁸³-4·10⁴¹-1 =: 99999999999999999999999999999999999999999599999999999999999999999999999999999999999_<83>; = 13 · 137 · 2969 · 966011 · 19576894984857248881997625912122528621643825060981473425904792886634081_<71>

10⁸⁵-4·10⁴²-1 =: 9999999999999999999999999999999999999999995999999999999999999999999999999999999999999_<85>; = 137 · 87037 · 19949644990069321_<17> · 42037835640022397405127233828635857223837222062393283928134651_<62>

10⁸⁷-4·10⁴³-1 =: 999999999999999999999999999999999999999999959999999999999999999999999999999999999999999_<87>; = 7 · 76018219698559_<14> · 128880567404353_<15> · 3486991762084429285438523_<25> · 4181632906804639258420048438806517_<34>

10⁸⁹-4·10⁴⁴-1 =: 99999999999999999999999999999999999999999999599999999999999999999999999999999999999999999_<89>; = 20935601 · 4776552629179358166025422437120386465141363727747772800981447821822741081089575599_<82>

10⁹¹-4·10⁴⁵-1 =: 9999999999999999999999999999999999999999999995999999999999999999999999999999999999999999999_<91>; = 13 · 479 · 25030471261_<11> · 553173690733_<12> · 97318941159199440533_<20> · 1191772105565888695309828426385501062443596153_<46>

10⁹³-4·10⁴⁶-1 =: 999999999999999999999999999999999999999999999959999999999999999999999999999999999999999999999_<93>; = 29261537 · 34174554808928867953860386759588192513605830068324845683943396411473532644577077410527_<86>

10⁹⁵-4·10⁴⁷-1 =: 99999999999999999999999999999999999999999999999599999999999999999999999999999999999999999999999_<95>; = 13 · 43 · 1801 · 7213 · 2974269059557322418797_<22> · 61936948866945854264869_<23> · 74752975888871388528161788533520804331629_<41>

10⁹⁷-4·10⁴⁸-1 =: 9999999999999999999999999999999999999999999999995999999999999999999999999999999999999999999999999_<97>; = 736163692210603204451_<21> · 13583935347274882563494726417155744680909144849612245764874721038411155776949_<77>

10⁹⁹-4·10⁴⁹-1 =: 999999999999999999999999999999999999999999999999959999999999999999999999999999999999999999999999999_<99>; = 7 · 113 · 137 · 9227901482923768305849566750025376729078040362840717192503252835272730628327811972279383945297_<94>

10¹⁰¹-4·10⁵⁰-1 =: 99999999999999999999999999999999999999999999999999599999999999999999999999999999999999999999999999999_<101>; = 137 · 1758560811461237726029_<22> · 170110589071199965225716045665378595373_<39> · 2440004871621207120718780266536678794831_<40>

10¹⁰³-4·10⁵¹-1 =: 9999999999999999999999999999999999999999999999999995999999999999999999999999999999999999999999999999999_<103>; = 13 · 31 · 61 · 6566643942114427521561609169_<28> · 61947195604635819770854250646250897150637806190952076410045507936140537_<71>

10¹⁰⁵-4·10⁵²-1 =: 999999999999999999999999999999999999999999999999999959999999999999999999999999999999999999999999999999999_<105>; = 67 · 14925373134328358208955223880597014925373134328358208358208955223880597014925373134328358208955223880597_<104>

10¹⁰⁷-4·10⁵³-1 =: 99999999999999999999999999999999999999999999999999999599999999999999999999999999999999999999999999999999999_<107>; = 13 · 47 · 1637 · 3559 · 374546534899059572140342925545382219_<36> · 75002603471220634662157651405579337049820078686199957972326517_<62> (Kenichiro Yamaguchi / GGNFS-0.77.0 / 0.92 hours on Pentium M 1.3GHz / May 29, 2005)

10¹⁰⁹-4·10⁵⁴-1 =: 9999999999999999999999999999999999999999999999999999995999999999999999999999999999999999999999999999999999999_<109>; = 163 · 173 · 1511 · 38971 · 3778126831_<10> · 697637286403_<12> · 828226457826773539_<18> · 39005950046315441269898777_<26> · 70725186403507264501279462608499_<32>

10¹¹¹-4·10⁵⁵-1 =: 999999999999999999999999999999999999999999999999999999959999999999999999999999999999999999999999999999999999999_<111>; = 7 · 1590646496312418081825900969362419_<34> · 89810742480071673480034617705907970811706653082603487540435533001853035752403_<77> (Kenichiro Yamaguchi / GGNFS-0.77.0 / 1.26 hours on Pentium M 1.3GHz / May 31, 2005)

10¹¹³-4·10⁵⁶-1 =: 99999999999999999999999999999999999999999999999999999999599999999999999999999999999999999999999999999999999999999_<113>; = 9283 · 92233 · 282363919 · 413633889443306629777357161514000460876645596909094077705711137549369107539589022982679384342339_<96>

10¹¹⁵-4·10⁵⁷-1 =: 9999999999999999999999999999999999999999999999999999999995999999999999999999999999999999999999999999999999999999999_<115>; = 13 · 137 · 1194883 · 4852747805592539495163744803133599_<34> · 968329084334870313982674682982929518043498846672500388409613165012933887_<72> (Kenichiro Yamaguchi / GGNFS-0.77.0 / 3.09 hours on Pentium M 1.3GHz / May 31, 2005)

10¹¹⁷-4·10⁵⁸-1 =: 999999999999999999999999999999999999999999999999999999999959999999999999999999999999999999999999999999999999999999999_<117>; = 137 · 7299270072992700729927007299270072992700729927007299270072700729927007299270072992700729927007299270072992700729927_<115>

10¹¹⁹-4·10⁵⁹-1 =: 99999999999999999999999999999999999999999999999999999999999599999999999999999999999999999999999999999999999999999999999_<119>; = 13 · 308393971 · 23793448646153_<14> · 60310927767427422968803_<23> · 17381904743309183789954408103368527186167885113743947830584157191922531707_<74>

10¹²¹-4·10⁶⁰-1 =: 9999999999999999999999999999999999999999999999999999999999995999999999999999999999999999999999999999999999999999999999999_<121>; = 480393481 · 4094317156436531656764569430473_<31> · 114616664758410776631617120793419_<33> · 44358172628315269260833164220111213839258758176717_<50> (Makoto Kamada / GMP-ECM 6.0.1 B1=3000000, sigma=955481116 for P31 / May 14, 2005) (Makoto Kamada / GGNFS-0.77.1 gnfs for P33 x P50 / 1.60 hours on Pentium 4 3.06GHz, Windows XP and Cygwin / May 17, 2005)

10¹²³-4·10⁶¹-1 =: 999999999999999999999999999999999999999999999999999999999999959999999999999999999999999999999999999999999999999999999999999_<123>; = 7 · 47 · 2251368737_<10> · 197336517199_<12> · 268046355481_<12> · 7330358007080381107_<19> · 553112759426567517799203632543_<30> · 6295079838882981527709998026784590476277_<40>

10¹²⁵-4·10⁶²-1 =: 99999999999999999999999999999999999999999999999999999999999999599999999999999999999999999999999999999999999999999999999999999_<125>; = 7653623 · 5621531592701_<13> · 3113842239924497_<16> · 2601070024525834540547695133_<28> · 286965460970540223894355002958897711729977697815694127298406913_<63>

10¹²⁷-4·10⁶³-1 =: 9999999999999999999999999999999999999999999999999999999999999995999999999999999999999999999999999999999999999999999999999999999_<127>; = 13 · 328543 · 600931 · 1159691297402406324541_<22> · 3359676199224728130532719425228453904361780139328315643281493364260416653063247947996194630691_<94>

10¹²⁹-4·10⁶⁴-1 =: 999999999999999999999999999999999999999999999999999999999999999959999999999999999999999999999999999999999999999999999999999999999_<129>; = 3023 · 34897 · 420865208494316969716449211_<27> · 4985055090497999350576552074159016833437_<40> · 4518151316303936463207381068077395206440013927504964047_<55> (Makoto Kamada / GMP-ECM 6.0 B1=3000000, sigma=3660474458 for P27 / May 26, 2005) (Kenichiro Yamaguchi / msieve.exe 0.88 for P40 x P55 / May 28, 2005)

10¹³¹-4·10⁶⁵-1 =: 99999999999999999999999999999999999999999999999999999999999999999599999999999999999999999999999999999999999999999999999999999999999_<131>; = 13 · 137 · 347 · 8623 · 67057 · 408048752821026243131_<21> · 50612544294970722084709792667_<29> · 3095535203928056850631948075200751_<34> · 4377217507217188817598381172409681_<34> (Makoto Kamada / GMP-ECM 6.0 B1=3000000, sigma=3035171655 for P29 / May 18, 2005) (Makoto Kamada / msieve 0.88 for P34(3095...) x P34(4377...) / May 18, 2005)

10¹³³-4·10⁶⁶-1 =: 9999999999999999999999999999999999999999999999999999999999999999995999999999999999999999999999999999999999999999999999999999999999999_<133>; = 31 · 137 · 7487097543124525709_<19> · 8115887942020483700339664052724822716727_<40> · 38749687619167420472280512971306132884586542208501397825133375024471019_<71> (Kenichiro Yamaguchi / GGNFS-0.77.1 / 17.31 hours on Pentium 4 2.4BGHz / Jun 19, 2005)

10¹³⁵-4·10⁶⁷-1 =: 999999999999999999999999999999999999999999999999999999999999999999959999999999999999999999999999999999999999999999999999999999999999999_<135>; = 7 · 199 · 776868191153854013968671748788920791_<36> · 4793208491666876855263351806795609063791229239_<46> · 192785880758047470136230491036835852228315194280607_<51> (Makoto Kamada / GMP-ECM 6.0 B1=3000000, sigma=4037378392 for P36 / Jun 5, 2005) (Sinkiti Sibata / GGNFS-0.77.1 gnfs / 16.97 hours / Jun 23, 2005)

10¹³⁷-4·10⁶⁸-1 =: 99999999999999999999999999999999999999999999999999999999999999999999599999999999999999999999999999999999999999999999999999999999999999999_<137>; = 43 · 647 · 21193 · 169603505994830115401634509352906604591693017355647186940222038431193824170549432236744961507661737469230383539350174987484490883_<129>

10¹³⁹-4·10⁶⁹-1 =: 9999999999999999999999999999999999999999999999999999999999999999999995999999999999999999999999999999999999999999999999999999999999999999999_<139>; = 13 · 3371 · 4459361 · 9093166788015632376354737529539617_<34> · 5627430068621602347797796257575031305382650518686424260015430403586239623261886314643346512049_<94> (Sinkiti Sibata / GGNFS-0.77.1 / 72.45 hours / Aug 8, 2005)

10¹⁴¹-4·10⁷⁰-1 =: 999999999999999999999999999999999999999999999999999999999999999999999959999999999999999999999999999999999999999999999999999999999999999999999_<141>; = 166609 · 168892705626840317_<18> · 69601747779002689794628246744089845171_<38> · 510587882640423485418476391438744702348606597088037480834827701746902682358566873_<81> (Sinkiti Sibata / GGNFS-0.77.1 gnfs / 161.09 hours for P38 x P81 / Aug 5, 2005)

10¹⁴³-4·10⁷¹-1 =: 99999999999999999999999999999999999999999999999999999999999999999999999599999999999999999999999999999999999999999999999999999999999999999999999_<143>; = 13 · 461 · 1593281 · 7476829 · 52605349 · 11206497793_<11> · 8007815567553387533_<19> · 194869174557877606595293994307427768604111_<42> · 1522610685152418997386812466952309432069612472477_<49> (Makoto Kamada / msieve 0.88 / 2.8 hours)

10¹⁴⁵-4·10⁷²-1 =: 9999999999999999999999999999999999999999999999999999999999999999999999995999999999999999999999999999999999999999999999999999999999999999999999999_<145>; = 65869487 · 165248121825939749137_<21> · 13229600165338069975361_<23> · 69443638555859011267751015980862364970490304305817364142593152436094142385631495472838954296161_<95>

10¹⁴⁷-4·10⁷³-1 =: 999999999999999999999999999999999999999999999999999999999999999999999999959999999999999999999999999999999999999999999999999999999999999999999999999_<147>; = 7 · 137 · 29063 · 3286768633530139_<16> · 25663232614150611623750872789_<29> · 2016779576750328637594482016979767_<34> · 210912355506634775150586992064179278706741771958204658151188871_<63> (Kenichiro Yamaguchi / msieve.exe 0.88 for P34 x P63 / 10:36:20 on Pentium4 2.4BGHz / May 28, 2005)

10¹⁴⁹-4·10⁷⁴-1 =: 99999999999999999999999999999999999999999999999999999999999999999999999999599999999999999999999999999999999999999999999999999999999999999999999999999_<149>; = 137 · 25471 · 79768751 · 152368553102117459_<18> · 91920304832658984824899_<23> · 25650382490615389836607447409347806115577051816973331809126578330683650197772489098514700057807_<95>

10¹⁵¹-4·10⁷⁵-1 =: 9999999999999999999999999999999999999999999999999999999999999999999999999995999999999999999999999999999999999999999999999999999999999999999999999999999_<151>; = 13 · 1399 · 16302156079139241779389_<23> · 140684559337675769430249526788079461538589821_<45> · 239743837024742116073004013726167157177219146951323324165616082687053255604169333_<81> (Sinkiti Sibata / GGNFS-0.77.1 / 96.94 hours / Aug 13, 2005)

Factorizations

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