160814

Ruby

x^3 + y^3 = z^3 ± 1(1)

x^3 + y^3 = z^3 を満たす自然数 (x, y, z) の組が存在しないことはよく知られているが、
Ramanujan's "Lost Notebook" の82頁には、
x^3 + y^3 = z^3 ± 1 を満たす自明でない自然数 (x, y, z) の組が載っている。
（ちなみに、(1, y, y) は自明な解である。）
Ramanujan の逸話で有名な
9^3 + 10^3 = 12^3 + 1
も解の一つに入っている。

ちなみに、オンライン整数列大辞典の

A272853、A272854、A272855

および

A051028、A051029、A051030

にも載っている。

def f(a, b, c, n)
  ary = [a]
  while ary.size < n + 1
    a, b, c = b, c, 82 * c + 82 * b - a
    ary << a
  end
  ary
end

def A272853(n)
  f(9, 791, 65601, n)
end

def A272854(n)
  f(10, 812, 67402, n)
end

def A272855(n)
  f(12, 1010, 83802, n)
end

def A051028(n)
  f(1, 135, 11161, n)
end

def A051029(n)
  f(2, 138, 11468, n)
end

def A051030(n)
  f(2, 172, 14258, n)
end

n = 30
a_ary = A272853(n)
b_ary = A272854(n)
c_ary = A272855(n)
(0..n).each{|i|
  a, b, c = a_ary[i], b_ary[i], c_ary[i]
  p [a, b, c, a * a * a + b * b * b == c * c * c + (-1) ** i]
}

a_ary = A051028(n)
b_ary = A051029(n)
c_ary = A051030(n)
(0..n).each{|i|
  a, b, c = a_ary[i], b_ary[i], c_ary[i]
  p [a, b, c, a * a * a + b * b * b == c * c * c + (-1) ** i]
}

出力結果
[9, 10, 12, true]
[791, 812, 1010, true]
[65601, 67402, 83802, true]
[5444135, 5593538, 6954572, true]
[451797561, 464196268, 577145658, true]
[37493753471, 38522696690, 47896135058, true]
[3111529740489, 3196919629018, 3974802064140, true]
[258219474707159, 265305806511788, 329860675188578, true]
[21429104870953665, 22017185020849402, 27374461238587818, true]
[1778357484814447079, 1827161050923988562, 2271750422127600332, true]
[147582242134728153849, 151632350041670201260, 188527910575352239722, true]
[12247547739697622322431, 12583657892407702716002, 15645544827332108296610, true]
[1016398880152767924607881, 1044291972719797655226922, 1298391692757989636378892, true]
[84348859504940040120131735, 86663650077850797681118508, 107750864954085807711151442, true]
[6999938940029870562046326081, 7192038664488896409877609258, 8942023399496364050389190778, true]
[580910583162974316609724933031, 596852545502500551222160449890, 742080191293244130374591683148, true]
[48208578463586838408045123115449, 49531569238043056855029439731628, 61583713853939766457040720510490, true]
[4000731101894544613551135493649279, 4110523394212071218416221337275218, 5110706169685707371804005210687538, true]
[332012472878783616086336200849774665, 341123910150363868071691341554111482, 424127028370059772093275391766555148, true]
[27553034517837145590552353535037647959, 28309174019085988978731965127653977772, 35197432648545275376370053511413389762, true]
[2286569852507604300399759007207275005889, 2349320319673986721366681414253726043610, 2920962782800887796466621166055544795082, true]
[189757744723613319787589445244668787840871, 194965277358921811884455825417931607641842, 242404713539825141831353186729098804602060, true]
[15747606242207397938069524196300302115786361, 16179768700470836399688466828274069708229292, 20116670261022685884205847877349145237175882, true]
[1306861560358490415539982918847680406822427135, 1342725836861720499362258290921329854175389378, 1669441226951343103247254020633249955880996162, true]
[108453761903512497091880512740161173464145665801, 111430064690822330610667749679642103826849089098, 138543505166700454883637877864682397192885505548, true]
[9000355376431178768210542574514529717117267834391, 9247352643501391720186060965119373287774299005740, 11497441487609186412238696608748005717053615964338, true]
[746921042481884325264383153171965805347269084588609, 767418839345924690444832392355228340781439968387338, 954149099966395771760928180648219792118257239534490, true]
[61985446170619967818175591170698647314106216753020199, 63686516313068247915200902504518832911571743077143298, 79182877855723239869744800297193494740098297265398348, true]
[5144045111118975444583309684014815761265468721416087865, 5285213435145318652271230075482707903319673235434506412, 6571224712925062513417057496486411843636040415788528378, true]
[426893758776704341932596528182059009537719797660782272639, 438609028600748379890596895362560237142621306797986888882, 545332468294924465373746027408074989527051256213182457042, true]
[35427037933355341404960928529426882975869477737123512541129, 36399264160426970212267271085017016974934248790997477270810, 45256023643765805563507503217373737718901618225278355406092, true]
[1, 2, 2, true]
[135, 138, 172, true]
[11161, 11468, 14258, true]
[926271, 951690, 1183258, true]
[76869289, 78978818, 98196140, true]
[6379224759, 6554290188, 8149096378, true]
[529398785665, 543927106802, 676276803218, true]
[43933719985479, 45139395574362, 56122825570732, true]
[3645969360009049, 3746025905565260, 4657518245567522, true]
[302571523160765631, 310875010766342202, 386517891556533610, true]
[25109790452983538281, 25798879867700837522, 32076327480946722092, true]
[2083810036074472911735, 2140996154008403172108, 2661948663027021400042, true]
[172931123203728268135681, 177676881902829762447458, 220909662703761829481378, true]
[14351199415873371782349831, 14745040201780861879966890, 18332840055749204825554348, true]
[1190976620394286129666900249, 1223660659865908706274804428, 1521404814964480238691529490, true]
[98836708293309875390570370879, 101549089728668641758928800618, 126258266801996110606571393338, true]
[8202255811724325371287673882665, 8427350786819631357284815646882, 10477914739750712700106734117548, true]
[680688395664825695941486361890359, 699368566216300734012880769890572, 869540665132507157998252360363162, true]
[56488934584368808437772080363017089, 58039163645166141291711819085270610, 72161397291258343401154839176024882, true]
[4687900882106946274639141183768528071, 4816551213982573426478068103307570042, 5988526434509309995137853399249702060, true]
[389039284280292171986610946172424812761, 399715711596908428256387940755443042892, 496975532666981471253040677298549246082, true]
[32285572694382143328614069391127490931135, 33171587511329416971853721014598464989978, 41242980684924952804007238362380337722762, true]
[2679313494349437604102981148517409322471401, 2752842047728744700235602456270917151125298, 3422670421316104101261347743400269481743148, true]
[222350734458308938997218821257553846274195191, 228452718373974480702583150149471525078409740, 284040401988551715451887855463859986646958538, true]
[18452431646545292499165059183228451831435729409, 18958822782992153153614165859949865664356883138, 23571930694628476278405430655756978622215815490, true]
[1531329475928800968491702693386703948162891345799, 1573353838269974737269273183225689378616542890698, 1956186207252174979392198856572365365657265727148, true]
[127081894070443935092312158491913199245688545971865, 130569409753624911040196060041872268559508703044812, 162339883271235894813274099664850568370930839537778, true]
[10546265878370917811693417452135408833443986424319039, 10835687655712597641599003710292172601060605809828682, 13472254125305327094522358073326024809421602415908442, true]
[875212986010715734435461336368747019976605184672508329, 899231506014391979341677111894208453619470773512735810, 1118034752517070912950542445986395208613622069680862892, true]
[72632131573011035040331597501153867249224786341393872311, 74625379311538821687717601283509009477815013595747243532, 92783412204791580447800500658797476290121210181095711610, true]
[6027591707573905192613087131259402234665680661151018893441, 6193007251351707808101219229419353578205026657673508477362, 7699905178245184106254491012234204136871446822961263200722, true]