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Positive integers k such that k^2 = A^2+B^2+C^2 and A^3+B^3+C^3 = m^3, where gcd(A,B,C) = 1 and A, B, C, m are positive integers.
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%I #37 Nov 13 2022 08:01:12

%S 75,119,551,755,4501,4895,16371,56863,61091,74201,201797,336709,

%T 534793,596827,879397,1007541

%N Positive integers k such that k^2 = A^2+B^2+C^2 and A^3+B^3+C^3 = m^3, where gcd(A,B,C) = 1 and A, B, C, m are positive integers.

%C From _Chai Wah Wu_, Sep 04 2020: (Start)

%C A. Martin and R. Davis showed that 91091088729334859 = sqrt(11868013975030087^2+16269106368215226^2+88837226814909894^2) is a term (see Links).

%C Table of values for k, A, B, C, m:

%C k A B C m

%C ---------------------------------------------

%C 75 14 23 70 71

%C 119 3 34 114 115

%C 551 18 349 426 493

%C 755 145 198 714 721

%C 4501 1016 2364 3693 4013

%C 4895 213 3450 3466 4357

%C 16371 3542 9286 13009 14497

%C 56863 6213 32194 46458 51157

%C 61091 29233 29574 44754 51985

%C 74201 32913 38444 54264 63185

%C 201797 106677 117252 124876 168373

%C 336709 110051 118044 295512 306467

%C 534793 116457 286752 436136 476393

%C 596827 202023 234550 510270 536023

%C 879397 43472 613560 628485 782597

%C 1007541 272267 417416 875656 914315

%C (End)

%H A. Martin and R. Davis, <a href="https://archive.org/details/bub_gb_UuFJAQAAIAAJ/page/n225/mode/2up">Solution of problem 143</a>, Jahrbuch über die Fortschritte der Mathematik, Band 29, Jahrgang 1898, pub. 1900, p. 157.

%H Ed Pegg Jr.'s Math Puzzles, <a href="http://www.mathpuzzle.com/cbumpkin.txt">A^2 + B^2 + C^2 = Square, A^3 + B^3 + C^3 = Cube</a>

%H Seiji Tomita, <a href="http://www.maroon.dti.ne.jp/fermat/dioph196e.html">A simultaneous equation {x^2+y^2+z^2=u^2, x^3+y^3+z^3=v^3} has infinitely many integer solutions</a>.

%e 56863 is in the sequence because 56863^2 = 6213^2 + 32194^2 + 46458^2, 6213^3 + 32194^3 + 46458^3 = 51157^3 and gcd(6213, 32194, 46458) = 1.

%Y Cf. A096910.

%K nonn,more

%O 1,1

%A _Mo Li_, Aug 21 2020