A337251 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.

75, 119, 551, 755, 4501, 4895, 16371, 56863, 61091, 74201, 201797, 336709, 534793, 596827, 879397, 1007541

Offset: 1

Author

Mo Li, Aug 21 2020

Keywords

• nonn
• more

Comments

From Chai Wah Wu, Sep 04 2020: (Start)

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

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

k A B C m

---------------------------------------------

75 14 23 70 71

119 3 34 114 115

551 18 349 426 493

755 145 198 714 721

4501 1016 2364 3693 4013

4895 213 3450 3466 4357

16371 3542 9286 13009 14497

56863 6213 32194 46458 51157

61091 29233 29574 44754 51985

74201 32913 38444 54264 63185

201797 106677 117252 124876 168373

336709 110051 118044 295512 306467

534793 116457 286752 436136 476393

596827 202023 234550 510270 536023

879397 43472 613560 628485 782597

1007541 272267 417416 875656 914315

(End)

Examples

			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.
		

Links

• A. Martin and R. Davis, Solution of problem 143, Jahrbuch über die Fortschritte der Mathematik, Band 29, Jahrgang 1898, pub. 1900, p. 157.
• Ed Pegg Jr.'s Math Puzzles, A^2 + B^2 + C^2 = Square, A^3 + B^3 + C^3 = Cube
• Seiji Tomita, A simultaneous equation {x^2+y^2+z^2=u^2, x^3+y^3+z^3=v^3} has infinitely many integer solutions.

Crossrefs

Cf. A096910.