The On-Line Encyclopedia of Integer Sequences (OEIS)

A255352 revision #23

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

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OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..56.

Mia Müßig, Table of n, a(n) for n = 1..120000 (terms 1..56 from M. F. Hasler)

Mia Müßig, Julia code for finding general taxicab numbers

EXAMPLE

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], ...

PROG

(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))]", "))))}

CROSSREFS

Cf. A018786, A255351, A366703.

Sequence in context: A033671 A122702 A142422 * A044391 A044772 A341659

Adjacent sequences: A255349 A255350 A255351 * A255353 A255354 A255355

KEYWORD

nonn

AUTHOR

M. F. Hasler, Feb 21 2015

STATUS

proposed