cp's OEIS Frontend

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.

%I A255352 #32 Mar 01 2025 01:26:19
%S A255352 59,158,133,134,7,239,157,227,193,292,256,257,118,316,266,268,177,474,
%T A255352 399,402,14,478,314,454,271,502,298,497,103,542,359,514,386,584,512,
%U A255352 514,222,631,503,558,236,632,532,536,21,717,471,681,295,790,665,670,579,876,768,771
%N 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.
%C A255352 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.
%C A255352 As far as is known, the existence of solutions to the equation with exponent n=5 remains an open question.
%C A255352 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).
%C A255352 See A366703 for the quadruples which consist only of prime numbers. - _Mia Muessig_, Oct 23 2023
%H A255352 Mia Müßig, <a href="/A255352/b255352.txt">Table of n, a(n) for n = 1..120000</a>
%H A255352 Mia Müßig, <a href="https://github.com/PhoenixSmaug/taxicab-numbers">Julia code for finding general taxicab numbers</a>
%e A255352 The quadruples [a,b,c,d] are, listed in order of increasing b = max{a,b,c,d}:
%e A255352 [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], ...
%o A255352 (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))]","))))}
%Y A255352 Cf. A018786, A255351, A366703.
%K A255352 nonn
%O A255352 1,1
%A A255352 _M. F. Hasler_, Feb 21 2015