A306981 Numbers that can be represented as sum of two fifth powers in two ways using Gaussian integers generated from Pell numbers.

244, 532344, 3368260700, 22612933199856, 152086272228543556, 1022930598444844458600, 6880230727163694887472044, 46276430783447828140913432544, 311255266556463822927417656862100, 2093502876579815396773605056889844056, 14080900036620070975438887843327087279356

Offset

1,1

Comments

Campbell and Zujev showed that if P(n) = A000129(n-1) are the Pell numbers, then a sequence of solutions to a^5 + b^5 = c^5 + d^5 can be generated using a = P(2n+3) + 1, b = P(2n+3) - 1, c = P(2n+3) + i*(P(2n+3) + P(2n+2)) and d = P(2n+3) - i*(P(2n+3) + P(2n+2)) (where i is the imaginary unit).

Links

Table of n, a(n) for n=1..11.

Geoffrey B. Campbell and Aleksander Zujev, Gaussian integer solutions for the fifth power taxicab number problem, arXiv:1511.07424 [math.NT], 2015.

Example

P(3) = 2 generates 244 = 3^5 + 1^5 = (2 + 3i)^5 + (2 - 3i)^5.

Mathematica

p[ n_] := With[ {s = Sqrt@2}, ((1 + s)^n - (1 - s)^n) / (2 s)] // Simplify; p0[n_] := p[n - 1]; Table[(p0[2n+3]-1)^5 + (p0[2n+3]+1)^5, {n, 0, 15}] (* after Michael Somos at A000129 *)

Crossrefs

Cf. A000129.

Sequence in context: A217471 A146552 A263948 * A164656 A291587 A220090

Adjacent sequences: A306978 A306979 A306980 * A306982 A306983 A306984

Keyword

nonn

Author

Amiram Eldar, Mar 18 2019

Status

approved