Taxicab numbers and their prime factor decomposition

EN
Refers to https://oeis.org/A011541 sequence.
See also http://www.christianboyer.com/taxicab/ and http://www.christianboyer.com/taxicab/TaxicabUpperBounds.pdf (or https://cs.uwaterloo.ca/journals/JIS/VOL11/Boyer/boyer-new.pdf)
Taxicab(n) - or T(n) or T(n, 1) - is the smallest number expressible as a sum of two positive cubes in n different ways.

FR
Le plus petit nombre T(n) qui est la somme de 2 cubes entiers positifs (x_i et y_i) de n façons, soit T(n) = x_i^3 + y_i^3 (pour i allant de 1 à n).
Ci-dessous, l'ensemble des solutions classées par ordre croissant des valeurs de x_ipuis y_i, et décroissant de n. On pourra également utiliser la notation T(n, k) pour la k-ième plus petite solution primitive pouvant s'écrire de n manières comme une somme de deux cubes d'entiers positifs.

Remarques

• Les valeurs x_i (sauf pour {x₁,y₁}) ont un rapport de 79 entre T(5) et T(6)
• Les valeurs x_i (sauf pour {x₃,y₃}) ont un rapport de 101 entre T(6) et T(7)
• Les valeurs x_i (sauf pour {x₈,y₈}) ont un rapport de 127 entre T(7) et T(8)
• Les valeurs x_i (sauf pour {x₂,y₂}) ont un rapport de 139 entre T(8) et T(9)
• Les valeurs x_i (sauf pour {x₂,y₂}) ont un rapport de 13 * 29 (377) entre T(9) et T(10)
• ...
• Les valeurs x_i (sauf pour {x₄,y₄}) ont un rapport de 3 * 61 (183) entre T(12) et T(13)
• Les valeurs x_i (sauf pour {x₆,y₆}) ont un rapport de 397 entre T(13) et T(14)
• ...
• La valeur de x₁ pour T(n+1) est supérieure (ou égale pour T(2)) à celle de y₁ pour T(n)
• Si on multiplie les x_i de T(5) par 79 ou 139 ou un multiple de 79 ou un multiple de 139, on obtient un multiple de T(6)
• Certains x_i ont une différence multiple de 11: {1,12}, {414,436}, {331954,365757}, {27093208,28906206}
  • de même pour: {255,167}, {13322,2421}, {231518,38787}, {26590452,582162}
• Sauf pour T(1), il y a plusieurs x_i multiples de 3 mais on peut constater un couple de x_i (espacés de 2 termes) multiples de 3 (et donc leur différence aussi): {9,12}, {228,423}, {10200,18072}, {221424,336588}, {1766742096,2685635652}

---

T(22) <= 4680247859298792255465896583920257161776853980641720488623945439873944558056759119053262369879441902130153472288312996410513410535472957059046536164424000000000 = 2^12 * 3^9 * 5^9 * 7^4 * 11^3 * 13^6 * 17^3 * 19^3 * 31^4 * 37^4 * 43 * 61^3 * 73 * 79^3 * 97^3 * 103^3 * 109^3 * 127^3 * 139^3 * 157 * 181^3 * 197^3 * 397^3 * 457^3 * 503^3 * 521^3 * 607^3 * 4261^3
T(21) <= 108017480260055891349349434078007118409816211596074421874782013152781209964290521967447172971431469705047741717793222992124981940703226072000000000 = 2^12 * 3^9 * 5^9 * 7^4 * 13^6 * 17^3 * 19^3 * 31 * 37^4 * 43 * 61^3 * 73 * 79^3 * 97^3 * 109^3 * 127^3 * 139^3 * 157 * 181^3 * 197^3 * 397^3 * 457^3 * 503^3 * 521^3 * 607^3 * 4261^3
T(20) <= 6897380753715950027554119762858815120976857451913869723850868496922922856256366585322070924221550029921862628606999892276808000000000 = 2^12 * 3^9 * 5^9 * 7^4 * 13^6 * 17^3 * 19^3 * 31 * 37^4 * 43 * 61^3 * 73 * 79^3 * 97^3 * 109^3 * 139^3 * 157 * 181^3 * 397^3 * 457^3 * 503^3 * 521^3 * 607^3 * 4261^3
T(19) <= 89155799373431424956779039683208074195259067508030416952016859893056008200986665364956362114476625742414288155368000000000 = 2^12 * 3^9 * 5^9 * 7^4 * 13^6 * 17^3 * 19^3 * 31 * 37^4 * 43 * 61^3 * 73 * 79^3 * 97^3 * 109^3 * 139^3 * 157 * 181^3 * 397^3 * 457^3 * 503^3 * 521^3 * 607^3
T(18) <= 32704115261272222010108668321371490956498583891096244871994833214348230808345220932546725287574509356813791744000 = 2^12 * 3^9 * 5^3 * 7^4 * 13^6 * 17^3 * 19^3 * 31 * 37^4 * 43 * 61^3 * 73 * 79^3 * 97^3 * 109^3 * 139^3 * 157 * 181^3 * 397^3 * 503^3 * 607^3 * 4261^3
T(17) <= 108883358434560363503260016942467566965657808716401027019007732065428469239764403684536752685040128000 = 2^12 * 3^9 * 5^3 * 7^4 * 13^6 * 17^3 * 19^3 * 31 * 37 * 43 * 61^3 * 73 * 79^3 * 97^3 * 109^3 * 139^3 * 157 * 397^3 * 503^3 * 607^3 * 4261^3
T(16) <= 1407430328457240141244921568479580896498768360005757901394557724136294559835243494681088000 = 2^12 * 3^9 * 5^3 * 7^4 * 13^6 * 17^3 * 19^3 * 31 * 37 * 43 * 61^3 * 73 * 79^3 * 97^3 * 109^3 * 139^3 * 157 * 397^3 * 503^3 * 607^3
T(15) <= 786630615595626829796137755567437844038832146564530660386728812791938555036352000 = 2^9 * 3^9 * 5^3 * 7^4 * 13^6 * 17^3 * 19^3 * 31 * 37 * 43 * 61^3 * 73 * 79^3 * 97^3 * 109^3 * 139^3 * 157 * 397^3 * 503^3

previous x_i * 503 and y_i * 503

(not complete)

x₁ = 10695065688795893596078797 = 3^3 * 7 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 397 * 503 * 733
x₁ = 28181600055013105932624120 = 2^3 * 3 * 5 * 7 * 13^2 * 17 * 23 * 61 * 79 * 97 * 109 * 139 * 359 * 397 * 503
x₁ = 56426766428922337436172912 = 2^4 * 3^5 * 7 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 397 * 503 * 3733
x₁ = 235180423863643682495328530 = 2 * 5 * 7 * 13^2 * 17 * 19^2 * 79 * 97 * 109 * 139 * 397 * 503 * 139721
x₁ = 264900495481098194325797280 = 2^5 * 3^4 * 5 * 7 * 13^2 * 19 * 61 * 79 * 97 * 109 * 139 * 397 * 503 * 643
x₁ = 271486889492468353693711800 = 2^3 * 3 * 5^2 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 503 * 1013 * 2297
x₁ = 288646131706313530332350160 = 2^4 * 3^4 * 5 * 7 * 17 * 19 * 61 * 79 * 109 * 139 * 397 * 503 * 1351243
x₁ = 426225140265221900183392812 = 2^2 * 3^4 * 7 * 11 * 13^2 * 17 * 19^2 * 61 * 79 * 97 * 109 * 397 * 503 * 1619
x₁ = 437199916596731902532072700 = 2^2 * 3^2 * 5^2 * 7 * 13 * 17 * 19 * 47 * 61 * 97 * 139 * 397^2 * 503 * 5393
x₁ = 442474705509905507209800120 = 2^3 * 3^3 * 5 * 7 * 13^2 * 17 * 19 * 61 * 97 * 101 * 109 * 139 * 397 * 503 * 593
x₁ = 622877571220997817580859640 = 2^3 * 3^3 * 5 * 7 * 13 * 17 * 19 * 61 * 79 * 97 * 139 * 397 * 503 * 1512283
x₁ = 628100814436347777756072660 = 2^2 * 3^5 * 5 * 7 * 13 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 397 * 503 * 3109
x₁ = 644639151547122830502047280 = 2^4 * 3^2 * 5 * 7 * 13^2 * 17 * 19 * 61 * 79 * 109 * 139 * 397 * 503 * 160709
x₁ = 659199020522444748146282040 = 2^3 * 3^2 * 5 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 397 * 503 * 23719

y₁ = 793795793421394953267595920 = 2^4 * 3^2 * 5 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 397 * 503 * 14281
y₁ = 803497501971159312701475000 =
y₁ = 813740846436226128138214860 =
y₁ = 816815317041515705690387520 =
y₁ = 887904512320529999991936240 =
y₁ = 889197010266142075310319780 =
y₁ = 891776614507430548042243188 =
y₁ = 913612658363701347420162360 =
y₁ = 915222810479227806552887280 =
y₁ = 915788065961417098709259000 =
y₁ = 918000867786830643403792750 =
y₁ = 923047140371302255936988088 =
y₁ = 923108668678583587617930240 =
y₁ = 923116945338403547262097203 = 3^4 * 7 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 397 * 503 * 21089

T(14) <= 6181115942163278484307116174514304039670628856329988877227399275142976000 = 2^9 * 3^9 * 5^3 * 7^4 * 13^6 * 17^3 * 19^3 * 31 * 37 * 43 * 61^3 * 73 * 79^3 * 97^3 * 109^3 * 139^3 * 157 * 397^3

previous x_i * 397 and y_i * 397

x₁ = 21262556041343724843099 = 3^3 * 7 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 397 * 733
x₂ = 56027037882729832868040 = 2^3 * 3 * 5 * 7 * 13^2 * 17 * 23 * 61 * 79 * 97 * 109 * 139 * 359 * 397
x₃ = 112180450156903255340304 = 2^4 * 3^5 * 7 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 397 * 3733
x₄ = 467555514639450661024510 = 2 * 5 * 7 * 13^2 * 17 * 19^2 * 79 * 97 * 109 * 139 * 397 * 139721
x₅ = 526641144097610724305760 = 2^5 * 3^4 * 5 * 7 * 13^2 * 19 * 61 * 79 * 97 * 109 * 139 * 397 * 643
x₆ = 539735366784231319470600 = 2^3 * 3 * 5^2 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 1013 * 2297
x₇ = 573849168402213777996720 = 2^4 * 3^4 * 5 * 7 * 17 * 19 * 61 * 79 * 109 * 139 * 397 * 1351243
x₈ = 847366084026286083863604 = 2^2 * 3^4 * 7 * 11 * 13^2 * 17 * 19^2 * 61 * 79 * 97 * 109 * 397 * 1619
x₉ = 869184724844397420540900 = 2^2 * 3^2 * 5^2 * 7 * 13 * 17 * 19 * 47 * 61 * 97 * 139 * 397^2 * 5393
x₁₀ = 879671382723470193260040 = 2^3 * 3^3 * 5 * 7 * 13^2 * 17 * 19 * 61 * 97 * 101 * 109 * 139 * 397 * 593
x₁₁ = 1238325191294230253639880 = 2^3 * 3^3 * 5 * 7 * 13 * 17 * 19 * 61 * 79 * 97 * 139 * 397 * 1512283
x₁₂ = 1248709372636874309654220 = 2^2 * 3^5 * 5 * 7 * 13 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 397 * 3109
x₁₃ = 1281588770471417158055760 = 2^4 * 3^2 * 5 * 7 * 13^2 * 17 * 19 * 61 * 79 * 109 * 139 * 397 * 160709
x₁₄ = 1310534832052574051980680 = 2^3 * 3^2 * 5 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 397 * 23719
y₁₄ = 1578122849744323962758640 = 2^4 * 3^2 * 5 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 397 * 14281
y₁₃ = 1597410540698129846325000 = 2^3 * 3^2 * 5^5 * 7 * 13^2 * 17 * 19 * 61 * 79 * 109 * 139 * 397 * 641
y₁₂ = 1617775042616751745801620 = 2^2 * 3^3 * 5 * 7 * 13 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 397 * 36251
y₁₁ = 1623887310221701204155840 = 2^6 * 3^4 * 5 * 7 * 13 * 17 * 19^2 * 61 * 79 * 97 * 139 * 397 * 4349
y₁₀ = 1765217718331073558632080 = 2^4 * 3^5 * 5 * 7 * 11 * 13^2 * 17 * 19 * 61 * 97 * 109 * 139 * 397 * 607
y₉ = 1767787296751773509563260 = 2^2 * 3^2 * 5 * 7 * 13 * 17 * 19 * 61 * 97 * 139 * 397 * 1023308809
y₈ = 1772915734607217789348396 = 2^2 * 3^3 * 7 * 13^2 * 17 * 19 * 61 * 79 * 83 * 97 * 109 * 397 * 25589
y₇ = 1816327352611732301034120 = 2^3 * 3^3 * 5 * 7 * 17 * 19 * 61 * 79 * 109 * 139 * 397 * 25661443
y₆ = 1819528450256914128335760 = 2^4 * 3 * 5 * 13^2 * 17 * 19^2 * 61 * 79 * 97 * 109 * 139 * 1032131
y₅ = 1820652218611167194253000 = 2^3 * 3^3 * 5^3 * 7 * 11 * 13^2 * 19 * 61 * 79 * 97^2 * 109 * 139 * 397
y₄ = 1825051427011591736389250 = 2 * 5^3 * 7 * 13^2 * 17 * 19 * 47 * 79 * 97 * 109 * 139 * 397 * 8819
y₃ = 1835083778074159554546696 = 2^3 * 3^3 * 7 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 397 * 1099181
y₂ = 1835206100752651267630080 = 2^10 * 3 * 5 * 7 * 13^2 * 17 * 61 * 79 * 97 * 109 * 139 * 397 * 2113
y₁ = 1835222555344738662548901 = 3^4 * 7 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 397 * 21089

T(13) <= 98785992977316717572070208794037178343163969803121800608526912000 = 2^9 * 3^9 * 5^3 * 7^4 * 13^6 * 17^3 * 19^3 * 31 * 37 * 43 * 61^3 * 73 * 79^3 * 97^3 * 109^3 * 139^3 * 157

previous x_i * 183 and y_i * 183

x₁ = 53558075670890994567 = 3^3 * 7 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 733
x₂ = 141126040006876153320 = 2^3 * 3 * 5 * 7 * 13^2 * 17 * 23 * 61 * 79 * 97 * 109 * 139 * 359
x₃ = 282570403417892330832 = 2^4 * 3^5 * 7 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 3733
x₄ = 1177721699343704435830 = 2 * 5 * 7 * 13^2 * 17 * 19^2 * 79 * 97 * 109 * 139 * 139721
x₅ = 1326552000245870842080 = 2^5 * 3^4 * 5 * 7 * 13^2 * 19 * 61 * 79 * 97 * 109 * 139 * 643
x₆ = 1445463900257465435760 = 2^4 * 3^4 * 5 * 7 * 17 * 19 * 61 * 79 * 109 * 139 * 1351243
x₇ = 2134423385456640009732 = 2^2 * 3^4 * 7 * 11 * 13^2 * 17 * 19^2 * 61 * 79 * 97 * 109 * 1619
x₈ = 2189382178449363779700 = 2^2 * 3^2 * 5^2 * 7 * 13 * 17 * 19 * 47 * 61 * 97 * 139 * 397 * 5393
x₉ = 2215796933812267489320 = 2^3 * 3^3 * 5 * 7 * 13^2 * 17 * 19 * 61 * 97 * 101 * 109 * 139 * 593
x₁₀ = 3119207030967834392040 = 2^3 * 3^3 * 5 * 7 * 13 * 17 * 19 * 61 * 79 * 97 * 139 * 1512283
x₁₁ = 3145363659034947883260 = 2^2 * 3^5 * 5 * 7 * 13 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 3109
x₁₂ = 3228183300935559592080 = 2^4 * 3^2 * 5 * 7 * 13^2 * 17 * 19 * 61 * 79 * 109 * 139 * 160709
x₁₃ = 3301095294842755798440 = 2^3 * 3^2 * 5 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 23719
y₁₃ = 3975120528323234163120 = 2^4 * 3^2 * 5 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 14281
y₁₂ = 4023704132740881225000 = 2^3 * 3^2 * 5^5 * 7 * 13^2 * 17 * 19 * 61 * 79 * 109 * 139 * 641
y₁₁ = 4075000107346981727460 = 2^2 * 3^3 * 5 * 7 * 13 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 36251
y₁₀ = 4090396247409826710720 = 2^6 * 3^4 * 5 * 7 * 13 * 17 * 19^2 * 61 * 79 * 97 * 139 * 4349
y₉ = 4446392237609757074640 = 2^4 * 3^5 * 5 * 7 * 11 * 13^2 * 17 * 19 * 61 * 97 * 109 * 139 * 607
y₈ = 4452864727334442089580 = 2^2 * 3^2 * 5 * 7 * 13 * 17 * 19 * 61 * 97 * 139 * 1023308809
y₇ = 4465782706819188386268 = 2^2 * 3^3 * 7 * 13^2 * 17 * 19 * 61 * 79 * 83 * 97 * 109 * 25589
y₆ = 4575131870558519649960 = 2^3 * 3^3 * 5 * 7 * 17 * 19 * 61 * 79 * 109 * 139 * 25661443
y₅ = 4586025739574728449000 = 2^3 * 3^3 * 5^3 * 7 * 11 * 13^2 * 19 * 61 * 79 * 97^2 * 109 * 139
y₄ = 4597106869046830570250 = 2 * 5^3 * 7 * 13^2 * 17 * 19 * 47 * 79 * 97 * 109 * 139 * 8819
y₃ = 4622377274745993840168 = 2^3 * 3^3 * 7 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 1099181
y₂ = 4622685392324058608640 = 2^10 * 3 * 5 * 7 * 13^2 * 17 * 61 * 79 * 97 * 109 * 139 * 2113
y₁ = 4622726839659291341433 = 3^4 * 7 * 13^2 * 17 * 19 * 61 * 79 * 97 * 109 * 139 * 21089

T(19) <= 20400824749409517528805616329601248054238975120047899653306832282155305931553802798551427446267572330642814901020208524571911529017539708401834884288000 = 2^9 × 3^12 × 5^3 × 7^4 × 11^3 × 13^4 × 17^3 × 19 × 29^3 × 41^3 × 43 × 73 × 79^3 × 97 × 101^3 × 127^3 × 139^3 × 157 × 271^3 × 349^3 × 601^3 × 727^3 × 733^3 × 2311^3 × 2971^3 × 4327^3 × 7549^3
T(18) <= 1669102760262770599073633207770749663711833995754118800009115774272350540676865796358235477762975575548685960795354954969975483624898697728 = 2^9 × 3^12 × 7^4 × 11^3 × 13^4 × 19 × 29^3 × 41^3 × 43 × 73 × 79^3 × 97 × 101^3 × 127^3 × 139^3 × 157 × 349^3 × 601^3 × 727^3 × 733^3 × 2311^3 × 2971^3 × 4327^3 × 7549^3
T(17) <= 390632494425592857308683115941747067399542361030496398592266116103976300384768055853952870509972688279894938439376270144482816 = 2^9 × 3^3 × 7^4 × 11^3 × 13^4 × 19 × 29^3 × 41^3 × 43 × 73 × 79^3 × 97 × 101^3 × 127^3 × 139^3 × 157 × 349^3 × 727^3 × 733^3 × 2311^3 × 2971^3 × 4327^3 × 7549^3
T(16) <= 133333879575067044255496407342541909372921060713622256889696224426242257299346008208169434186136206003482924680704 = 2^9 × 3^3 × 7^4 × 11^3 × 13^4 × 19 × 29^3 × 43 × 73 × 79^3 × 97 × 101^3 × 127^3 × 139^3 × 157 × 727^3 × 733^3 × 2311^3 × 2971^3 × 4327^3 × 7549^3
T(15) <= 254361007450418467683691805874871224061387429621450309607817376336263717732850126485501423293412997632 = 2^9 × 3^3 × 7^4 × 13^4 × 19 × 29^3 × 43 × 73 × 79^3 × 97 × 101^3 × 127^3 × 139^3 × 157 × 727^3 × 2311^3 × 2971^3 × 4327^3
T(14) <= 591265120715306076227178149379165201865336472346517495072293984450950965960475614042157568 = 2^9 × 3^3 × 7^4 × 13^4 × 19 × 29^3 × 43 × 73 × 79^3 × 97 × 101^3 × 127^3 × 139^3 × 157 × 727^3 × 2311^3 × 2971^3 × 4327^3
T(13) <= 5988146776742829080553965820313279739849705084894534523771076163371248442670016 = 2^6 × 3^3 × 7^4 × 13^4 × 19 × 29^3 × 43 × 73 × 79^3 × 97 × 101^3 × 127^3 × 139^3 × 157 × 727^3 × 2971^3 × 4327^3

T(12) <= 16119148654034302034428760115512552827992287460693283776000 = 2^9 * 3^6 * 5^3 * 7^4 * 13^6 * 17^3 * 19^3 * 31 * 37 * 43 * 73 * 79^3 * 97^3 * 109^3 * 139^3 * 157
x₁ = 292667080168803249 = 3^2 * 7 * 13^2 * 17 * 19 * 79 * 97 * 109 * 139 * 733
x₂ = 771180546485662040 = 2^3 * 5 * 7 * 13^2 * 17 * 23 * 79 * 97 * 109 * 139 * 359
x₃ = 1544100565125094704 = 2^4 * 3^4 * 7 * 13^2 * 17 * 19 * 79 * 97 * 109 * 3733
x₄ = 7248918034130441760 = 2^5 * 3^3 * 5 * 7 * 13^2 * 19 * 79 * 97 * 109 * 139 * 643
x₅ = 7898709837472488720 = 2^4 * 3^3 * 5 * 7 * 17 * 19 * 79 * 109 * 139 * 1351243
x₆ = 11663515767522623004 = 2^2 * 3^3 * 7 * 11 * 13^2 * 17 * 19^2 * 79 * 97 * 109 * 1619
x₇ = 11963837040706905900 = 2^2 * 3 * 5^2 * 7 * 13 * 17 * 19 * 47 * 97 * 139 * 397 * 5393
x₈ = 12108179966187254040 = 2^3 * 3^2 * 5 * 7 * 13^2 * 17 * 19 * 97 * 101 * 109 * 139 * 593
x₉ = 17044847163758657880 = 2^3 * 3^2 * 5 * 7 * 13 * 17 * 19 * 79 * 97 * 139 * 1512283
x₁₀ = 17187779557568021220 = 2^2 * 3^4 * 5 * 7 * 13 * 17 * 19 * 79 * 97 * 109 * 139 * 3109
x₁₁ = 17640345906751691760 = 2^4 * 3 * 5 * 7 * 13^2 * 17 * 19 * 79 * 109 * 139 * 160709
x₁₂ = 18038772102965878680 = 2^3 * 3 * 5 * 13^2 * 17 * 19 * 79 * 97 * 109 * 139 * 23719
y₁₂ = 21721970100126962640 = 2^4 * 3 * 5 * 13^2 * 17 * 19 * 79 * 97 * 109 * 139 * 14281
y₁₁ = 21987454277272575000 = 2^3 * 3 * 5^5 * 7 * 13^2 * 17 * 19 * 79 * 109 * 139 * 641
y₁₀ = 22267760149437058620 = 2^2 * 3^2 * 5 * 7 * 13 * 17 * 19 * 79 * 97 * 109 * 139 * 36251
y₉ = 22351892062348779840 = 2^6 * 3^3 * 5 * 7 * 13 * 17 * 19^2 * 79 * 97 * 139 * 4349
y₈ = 24297225342129820080 = 2^4 * 3^4 * 5 * 7 * 11 * 13^2 * 17 * 19 * 97 * 109 * 139 * 607
y₇ = 24332594138439574260 = 2^2 * 3 * 5 * 7 * 13 * 17 * 19 * 97 * 139 * 1023308809
y₆ = 24403184190268788996 = 2^2 * 3^2 * 7 * 13^2 * 17 * 19 * 79 * 83 * 97 * 109 * 25589
y₅ = 25000720604144916120 = 2^3 * 3^2 * 5 * 7 * 17 * 19 * 79 * 109 * 139 * 25661443
y₄ = 25060249943031303000 = 2^3 * 3^2 * 5^3 * 7 * 11 * 13^2 * 19 * 79 * 97^2 * 109 * 139
y₃ = 25258892211726742296 = 2^3 * 3^2 * 7 * 13^2 * 17 * 19 * 79 * 97 * 109 * 1099181
y₂ = 25260575914339118080 = 2^10 * 5 * 7 * 13^2 * 17 * 79 * 97 * 109 * 139 * 2113
y₁ = 25260802402509788751 = 3^3 * 7 * 13^2 * 17 * 19 * 79 * 97 * 109 * 139 * 21089

• x₁₂ et y₁₂ ne contiennent pas 7
• x₅ et y₅ ne contiennent pas 13
• x₄ et y₄ ne contiennent pas 17
• x₂ et y₂ ne contiennent pas 19

T(11) <= 87039729655193781808322993393446581825405320183232000 = 2^9 * 3^3 * 5^3 * 7^4 * 13^6 * 17^3 * 31 * 37 * 43 * 73 * 79^3 * 97^3 * 109^3 * 139^3 * 157
x₁ = 5134510178400057 = 3 * 7 * 13^2 * 17 * 79 * 97 * 109 * 139 * 733
x₂ = 27089483598685872 = 2^4 * 3^3 * 7 * 13^2 * 17 * 79 * 97 * 109 * 3733
x₃ = 127174000598779680 = 2^5 * 3^2 * 5 * 7 * 13^2 * 79 * 97 * 109 * 139 * 643
x₄ = 138573856797762960 = 2^4 * 3^2 * 5 * 7 * 17 * 79 * 109 * 139 * 1351243
x₅ = 204623083640747772 = 2^2 * 3^2 * 7 * 11 * 13^2 * 17 * 19 * 79 * 97 * 109 * 1619
x₆ = 209891877907138700 = 2^2 * 5^2 * 7 * 13 * 17 * 47 * 97 * 139 * 397 * 5393
x₇ = 212424209933109720 = 2^3 * 3 * 5 * 7 * 13^2 * 17 * 97 * 101 * 109 * 139 * 593
x₈ = 299032406381730840 = 2^3 * 3 * 5 * 7 * 13 * 17 * 79 * 97 * 139 * 1512283
x₉ = 301539992238035460 = 2^2 * 3^3 * 5 * 7 * 13 * 17 * 79 * 97 * 109 * 139 * 3109
x₁₀ = 309479752750029680 = 2^4 * 5 * 7 * 13^2 * 17 * 79 * 109 * 139 * 160709
x₁₁ = 316469686016945240 = 2^3 * 5 * 13^2 * 17 * 79 * 97 * 109 * 139 * 23719
y₁₁ = 381087194739069520 = 2^4 * 5 * 13^2 * 17 * 79 * 97 * 109 * 139 * 14281
y₁₀ = 385744811881975000 = 2^3 * 5^5 * 7 * 13^2 * 17 * 79 * 109 * 139 * 641
y₉ = 390662458762053660 = 2^2 * 3 * 5 * 7 * 13 * 17 * 79 * 97 * 109 * 139 * 36251
y₈ = 392138457234189120 = 2^6 * 3^2 * 5 * 7 * 13 * 17 * 19 * 79 * 97 * 139 * 4349
y₇ = 426267111265435440 = 2^4 * 3^3 * 5 * 7 * 11 * 13^2 * 17 * 97 * 109 * 139 * 607
y₆ = 426887616463852180 = 2^2 * 5 * 7 * 13 * 17 * 97 * 139 * 1023308809
y₅ = 428126038425768228 = 2^2 * 3 * 7 * 13^2 * 17 * 79 * 83 * 97 * 109 * 25589
y₄ = 438609133406051160 = 2^3 * 3 * 5 * 7 * 17 * 79 * 109 * 139 * 25661443
y₃ = 439653507772479000 = 2^3 * 3 * 5^3 * 7 * 11 * 13^2 * 79 * 97^2 * 109 * 139
y₂ = 443138459854855128 = 2^3 * 3 * 7 * 13^2 * 17 * 79 * 97 * 109 * 1099181
y₁ = 443171971973855943 = 3^2 * 7 * 13^2 * 17 * 79 * 97 * 109 * 139 * 21089

• x₁₁ et y₁₁ ne contiennent pas 7
• x₄ et y₄ ne contiennent pas 13
• x₃ et y₃ ne contiennent pas 17

T(10) <= 7335345315241855602572782233444632535674275447104 = 2^6 * 3^3 * 7^4 * 13^4 * 19 * 29^3 * 43 * 73 * 79^3 * 97 * 101^3 * 127^3 * 139^3 * 157
x₁ = 391313741613522 = 2 * 3 * 7 * 13 * 29 * 83 * 101 * 127 * 139 * 167
x₂ = 904069333568884 = 2^2 * 7 * 79 * 101 * 139 * 29112563
x₃ = 1536982932706676 = 2^2 * 7 * 13 * 29 * 79 * 101 * 127 * 143687
x₄ = 2059655218961613 = 3 * 7 * 13 * 29 * 79 * 101 * 127 * 139 * 1847
x₅ = 3058262831974168 = 2^3 * 7 * 13 * 29 * 127 * 139 * 8205913
x₆ = 5726433061530961 = 13 * 29 * 79 * 101 * 127 * 139 * 107839
x₇ = 10901351979041108 = 2^2 * 13 * 17 * 29 * 79 * 101 * 127 * 139 * 3019
x₈ = 11757988429199376 = 2^4 * 3 * 7 * 13 * 29 * 79 * 101 * 127 * 139 * 659
x₉ = 12293996879974082 = 2 * 7 * 13 * 23 * 29 * 79 * 101 * 127 * 139 * 719
x₁₀ = 15137846555691028 = 2^2 * 13 * 29 * 79 * 101 * 139 * 9051061
y₁₀ = 15695330667573128 = 2^3 * 11 * 13 * 29 * 79 * 101 * 139 * 426563
y₉ = 17627318136364846 = 2 * 7 * 13 * 29 * 79 * 101 * 127 * 131 * 139 * 181
y₈ = 17873391364113012 = 2^2 * 3 * 7 * 13 * 29 * 79 * 101 * 127 * 139 * 4007
y₇ = 18211330514175448 = 2^3 * 13 * 29 * 79 * 101 * 127 * 139 * 163 * 263
y₆ = 19262797062004847 = 13 * 29 * 79 * 101 * 127 * 139 * 362753
y₅ = 19404743826965588 = 2^2 * 7 * 13 * 29 * 127 * 139 * 191 * 457 * 1193
y₄ = 19422314536358643 = 3 * 7 * 13 * 29 * 79 * 101 * 127 * 139 * 17417
y₃ = 19426825887781312 = 2^6 * 7 * 11 * 13 * 17 * 29 * 79 * 101 * 127 * 607
y₂ = 19429379778270560 = 2^5 * 5 * 7 * 79 * 101 * 139 * 15641473
y₁ = 19429979328281886 = 2 * 3 * 7 * 13 * 29 * 101 * 127 * 139 * 311 * 2213

• x₂ et y₂ ne contiennent pas 13 * 29
• x₃ et y₃ ne contiennent pas 139
• x₅ et y₅ ne contiennent pas 101

T(9) <= 136897813798023990395783317207361432493888 = 2^6 * 3^3 * 7^4 * 13 * 19 * 43 * 73 * 79^3 * 97 * 101^3 * 127^3 * 139^3 * 157
x₁ = 1037967484386 = 2 * 3 * 7 * 83 * 101 * 127 * 139 * 167
x₂ = 4076877805588 = 2^2 * 7 * 79 * 101 * 127 * 143687
x₃ = 5463276442869 = 3 * 7 * 79 * 101 * 127 * 139 * 1847
x₄ = 8112103002584 = 2^3 * 7 * 127 * 139 * 8205913
x₅ = 15189477616793 = 79 * 101 * 127 * 139 * 107839
x₆ = 28916052994804 = 2^2 * 17 * 79 * 101 * 127 * 139 * 3019
x₇ = 31188298220688 = 2^4 * 3 * 7 * 79 * 101 * 127 * 139 * 659
x₈ = 32610071299666 = 2 * 7 * 23 * 79 * 101 * 127 * 139 * 719
x₉ = 40153439139764 = 2^2 * 79 * 101 * 139 * 9051061
y₉ = 41632176837064 = 2^3 * 11 * 79 * 101 * 139 * 426563
y₈ = 46756812032798 = 2 * 7 * 79 * 101 * 127 * 131 * 139 * 181
y₇ = 47409526164756 = 2^2 * 3 * 7 * 79 * 101 * 127 * 139 * 4007
y₆ = 48305916483224 = 2^3 * 79 * 101 * 127 * 139 * 163 * 263
y₅ = 51094952419111 = 79 * 101 * 127 * 139 * 362753
y₄ = 51471469037044 = 2^2 * 7 * 127 * 139 * 191 * 457 * 1193
y₃ = 51518075693259 = 3 * 7 * 79 * 101 * 127 * 139 * 17417
y₂ = 51530042142656 = 2^6 * 7 * 11 * 17 * 79 * 101 * 127 * 607
y₁ = 51538406706318 = 2 * 3 * 7 * 101 * 127 * 139 * 311 * 2213

T(8) <= 50974398750539071400590819921724352 = 2^6 * 3^3 * 7^4 * 13 * 19 * 43 * 73 * 79^3 * 97 * 101^3 * 127^3 * 157
x₁ = 7467391974 = 2 * 3 * 7 * 83 * 101 * 127 * 167
x₂ = 39304147071 = 3 * 7 * 79 * 101 * 127 * 1847
x₃ = 58360453256 = 2^3 * 7 * 127 * 8205913
x₄ = 109276817387 = 79 * 101 * 127 * 107839
x₅ = 208029158236 = 2^2 * 17 * 79 * 101 * 127 * 3019
x₆ = 224376246192 = 2^4 * 3 * 7 * 79 * 101 * 127 * 659
x₇ = 234604829494 = 2 * 7 * 23 * 79 * 101 * 127 * 719
x₈ = 288873662876 = 2^2 * 79 * 101 * 9051061
y₈ = 299512063576 = 2^3 * 11 * 79 * 101 * 426563
y₇ = 336379942682 = 2 * 7 * 79 * 101 * 127 * 131 * 181
y₆ = 341075727804 = 2^2 * 3 * 7 * 79 * 101 * 127 * 4007
y₅ = 347524579016 = 2^3 * 79 * 101 * 127 * 163 * 263
y₄ = 367589585749 = 79 * 101 * 127 * 362753
y₃ = 370298338396 = 2^2 * 7 * 127 * 191 * 457 * 1193
y₂ = 370633638081 = 3 * 7 * 79 * 101 * 127 * 17417
y₁ = 370779904362 = 2 * 3 * 7 * 101 * 127 * 311 * 2213

T(7) <= 24885189317885898975235988544 = 2^6 * 3^3 * 7^4 * 13 * 19 * 43 * 73 * 79^3 * 97 * 101^3 * 157
x₁ = 58798362 = 2 * 3 * 7 * 83 * 101 * 167
x₂ = 309481473 = 3 * 7 * 79 * 101 * 1847
x₃ = 459531128 = 2^3 * 7 * 8205913
x₄ = 860447381 = 79 * 101 * 107839
x₅ = 1638024868 = 2^2 * 17 * 79 * 101 * 3019
x₆ = 1766742096 = 2^4 * 3 * 7 * 79 * 101 * 659
x₇ = 1847282122 = 2 * 7 * 23 * 79 * 101 * 719
y₇ = 2648660966 = 2 * 7 * 79 * 101 * 131 * 181
y₆ = 2685635652 = 2^2 * 3 * 7 * 79 * 101 * 4007
y₅ = 2736414008 = 2^3 * 79 * 101 * 163 * 263
y₄ = 2894406187 = 79 * 101 * 362753
y₃ = 2915734948 = 2^2 * 7 * 191 * 457 * 1193
y₂ = 2918375103 = 3 * 7 * 79 * 101 * 17417
y₁ = 2919526806 = 2 * 3 * 7 * 101 * 311 * 2213

T(6) = 24153319581254312065344 = 2^6 * 3^3 * 7^4 * 13 * 19 * 43 * 73 * 79^3 * 97 * 157
x₁ = 582162 = 2 * 3 * 7 * 83 * 167
x₂ = 3064173 = 3 * 7 * 79 * 1847
x₃ = 8519281 = 79 * 107839
x₄ = 16218068 = 2^2 * 17 * 79 * 3019
x₅ = 17492496 = 2^4 * 3 * 7 * 79 * 659
x₆ = 18289922 = 2 * 7 * 23 * 79 * 719
y₆ = 26224366 = 2 * 7 * 79 * 131 * 181
y₅ = 26590452 = 2^2 * 3 * 7 * 79 * 4007
y₄ = 27093208 = 2^3 * 79 * 163 * 263
y₃ = 28657487 = 79 * 362753
y₂ = 28894803 = 3 * 7 * 79 * 17417
y₁ = 28906206 = 2 * 3 * 7 * 311 * 2213

T(5) = 48988659276962496 = 2^6 * 3^3 * 7^4 * 13 * 19 * 43 * 73 * 97 * 157
x₁ = 38787 = 3 * 7 * 1847
x₂ = 107839 = 107839
x₃ = 205292 = 2^2 * 17 * 3019
x₄ = 221424 = 2^4 * 3 * 7 * 659
x₅ = 231518 = 2 * 7 * 23 * 719
y₅ = 331954 = 2 * 7 * 131 * 181
y₄ = 336588 = 2^2 * 3 * 7 * 4007
y₃ = 342952 = 2^3 * 163 * 263
y₂ = 362753 = 362753
y₁ = 365757 = 3 * 7 * 17417

T(4) = 6963472309248 = 2^10 * 3^3 * 7 * 13 * 19 * 31 * 37 * 127
x₁ = 2421 = 3^2 * 269
x₂ = 5436 = 2^2 * 3^2 * 151
x₃ = 10200 = 2^3 * 3 * 5^2 * 17
x₄ = 13322 = 2 * 6661
y₄ = 16630 = 2 * 5 * 1663
y₃ = 18072 = 2^3 * 3^2 * 251
y₂ = 18948 = 2^2 * 3 * 1579
y₁ = 19083 = 3 * 6361

T(3) = 87539319 = 3^3 * 7 * 31 * 67 * 223
x₁ = 167 = 167
x₂ = 228 = 2^2 * 3 * 19
x₃ = 255 = 3 * 5 * 17
y₃ = 414 = 2 * 3^2 * 23
y₂ = 423 = 3^2 * 47
y₁ = 436 = 2^2 * 109

T(2) = 1729 = 7 * 13 * 19
x₁ = 1
x₂ = 9 = 3^2
y₂ = 10 = 2 * 5
y₁ = 12 = 2^2 * 3

T(1) = 2
x₁ = 1
y₁ = 1

---

T(7, ?) = 64866613980488624314617087936
x₁ = 80920518 = 2 * 3 * 7 * 83 * 139 * 167
x₂ = 317835644 = 2^2 * 7 * 79 * 143687
x₃ = 425920047 = 3 * 7 * 79 * 139 * 1847
x₄ = 1184180059 = 79 * 139 * 107839
x₅ = 2254311452 = 2^2 * 17 * 79 * 139 * 3019
x₆ = 2431456944 = 2^4 * 3 * 7 * 79 * 139 * 659
x₇ = 2542299158 = 2 * 7 * 23 * 79 * 139 * 719
y₇ = 3645186874 = 2 * 7 * 79 * 131 * 139 * 181
y₆ = 3696072828 = 2^2 * 3 * 7 * 79 * 139 * 4007
y₅ = 3765955912 = 2^3 * 79 * 139 * 163 * 263
y₄ = 3983390693 = 79 * 139 * 362753
y₃ = 4016377617 = 3 * 7 * 79 * 139 * 17417
y₂ = 4017310528 = 2^6 * 7 * 11 * 17 * 79 * 607
y₁ = 4017962634 = 2 * 3 * 7 * 139 * 311 * 2213

T(4, ?) = 1801049058342701083 = 7 * 31 * 37 * 43 * 163 * 193 * 9151 * 18121
x₁ = 92227
x₂ = 136635 = 3 * 5 * 9109
x₃ = 341995 = 5 * 68399
x₄ = 600259 = 11 * 197 * 277
y₄ = 1165884 = 2^2 * 3 * 97157
y₃ = 1207602 = 2 * 3^3 * 11 * 19 * 107
y₂ = 1216102 = 2 * 23 * 26437
y₁ = 1216500 = 2^2 * 3 * 5^3 * 811

T(3, ?) = 15170835645 = 3^2 * 5 * 7 * 31 * 37 * 199 * 211
x₁ = 517 = 11 * 47
x₂ = 709
x₃ = 1733
y₃ = 2152 = 2^3 * 269
y₂ = 2456 = 2^3 * 307
y₁ = 2468 = 2^2 * 617

T(3, 4) = 175959000 = 35^3 ∗ T(2, 2)
x₁ = 70 = 2 * 5 * 7
x₂ = 198 = 2 * 3^2 * 11
x₃ = 315 = 3^2 * 5 * 7
y₃ = 525 = 3 * 5^2 * 7
y₂ = 552 = 2^3 * 3 * 23
y₁ = 560 = 2^4 * 5 * 7

T(2, 2) = 4104
x₁ = 2
x₂ = 9 = 3^2
y₂ = 15 = 3 * 5
y₁ = 16 = 2^4