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A306981
Numbers that can be represented as sum of two fifth powers in two ways using Gaussian integers generated from Pell numbers.
0
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
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
KEYWORD
nonn
AUTHOR
Amiram Eldar, Mar 18 2019
STATUS
approved