A255352 List of quadruples (a,b,c,d) with a^4 + b^4 = c^4 + d^4, a < c < d < b, listed in order of the largest term b.

59, 158, 133, 134, 7, 239, 157, 227, 193, 292, 256, 257, 118, 316, 266, 268, 177, 474, 399, 402, 14, 478, 314, 454, 271, 502, 298, 497, 103, 542, 359, 514, 386, 584, 512, 514, 222, 631, 503, 558, 236, 632, 532, 536, 21, 717, 471, 681, 295, 790, 665, 670, 579, 876, 768, 771

Offset: 1

Author

M. F. Hasler, Feb 21 2015

Keywords

• nonn

Comments

The Ramanujan taxicab number 1729 = 1^3 + 12^3 = 9^3 + 10^3 satisfies the equation a^n + b^n = c^n + d^n for n=3. The present sequence corresponds to the same equation with exponent n=4.

As far as is known, the existence of solutions to the equation with exponent n=5 remains an open question.

See A018786 for the values of a^4 + b^4 = c^4 + d^4. See A255351 for the list of b-values, which are sufficient to reconstruct the quadruples (cf. inner loops of the PARI code).

See A366703 for the quadruples which consist only of prime numbers. - Mia Muessig, Oct 23 2023

Examples

			The quadruples [a,b,c,d] are, listed in order of increasing b = max{a,b,c,d}:
[59, 158, 133, 134], [7, 239, 157, 227], [193, 292, 256, 257], [118, 316, 266, 268], [177, 474, 399, 402], [14, 478, 314, 454], [271, 502, 298, 497], [103, 542, 359, 514], [386, 584, 512, 514], [222, 631, 503, 558], [236, 632, 532, 536], [21, 717, 471, 681], [295, 790, 665, 670], [579, 876, 768, 771], [354, 948, 798, 804], [28, 956, 628, 908], ...
		

Links

• Mia Müßig, Table of n, a(n) for n = 1..120000
• Mia Müßig, Julia code for finding general taxicab numbers

Crossrefs

Cf. A018786, A255351, A366703.

Programs

PARI

{n=4;for(b=1,999,for(a=1,b,t=a^n+b^n;for(c=a+1,sqrtn(t\2,n),ispower(t-c^n,n)||next;print1([a,b,c,round(sqrtn(t-c^n,n))]","))))}