A020896 Positive numbers k such that k = x^5 + y^5 has a solution in nonzero integers x, y.

2, 31, 33, 64, 211, 242, 244, 275, 486, 781, 992, 1023, 1025, 1056, 1267, 2048, 2101, 2882, 3093, 3124, 3126, 3157, 3368, 4149, 4651, 6250, 6752, 7533, 7744, 7775, 7777, 7808, 8019, 8800, 9031, 10901, 13682, 15552, 15783, 15961, 16564

Offset: 0

Author

Steven Finch

Keywords

• nonn
• nice

Comments

68101 = (15/2)^5 + (17/2)^5 is believed to be the smallest positive integer k which is the sum of two nonzero fifth powers of rational numbers but not the sum of two nonzero fifth powers of integers.

Examples

			31 = 2^5 + (-1)^5.
		

References

• J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 99.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..5000
• Steven R. Finch, On a Generalized Fermat-Wiles Equation [broken link]
• Steven R. Finch, On a Generalized Fermat-Wiles Equation [From the Wayback Machine]

Crossrefs

Cf. A001481, A020897, A003336.

Programs

Mathematica

Select[Union[Total/@(Select[Tuples[Range[-8,8],{2}], !MemberQ[#, 0]&]^5)],#>0&]  (* Harvey P. Dale, Apr 03 2011 *)

Formula

See Theorem 3.5.6 of J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 99.