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A255352
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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.
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4
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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
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OFFSET
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1,1
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COMMENTS
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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
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LINKS
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EXAMPLE
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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], ...
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PROG
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(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))]", "))))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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