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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Mar 18 2019
STATUS
approved