Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #32 Jan 26 2022 21:03:50
%S 1,16,61,81,256,625,976,1296,2401,4096,4941,6561,10000,14641,15616,
%T 20736,28561,38125,38416,50625,65536,79056,83521,104976,130321,146461,
%U 160000,194041,194481,229981,234256,249856,279841,331776,390625,400221,456976,531441
%N Positive integers representable by the two cyclotomic binary forms Phi_5(x,y) and Phi_12(u,v).
%C Positive integers C such that Phi_5(x,y) = Phi_12(u,v) = C has a solution with nonzero (x,y,u,v).
%C A cyclotomic binary form over Z is a homogeneous polynomial in two variables which has the form f(x, y) = y^EulerPhi(k)*CyclotomicPolynomial(k, x/y) where k is some integer >= 3. An integer n is represented by f if f(x,y) = n has an integer solution.
%H Étienne Fouvry, Claude Levesque and Michel Waldschmidt, <a href="https://arxiv.org/abs/1712.09019">Representation of integers by cyclotomic binary forms</a>, arXiv:1712.09019 [math.NT], 2017.
%e Phi_5(1,-3) = 1^4 + 1^3*(-3) + 1^2*(-3)^2 + 1*(-3)^3 + (-3)^4 = 1 - 3 + 9 - 27 + 81 = 61 and Phi_12(2, 3) = 2^4 - 2^2*3^2 + 3^4 = 16 - 36 + 81 = 61, so 61 is a term.
%Y Cf. A296095.
%K nonn
%O 1,2
%A _Shashi Kant Pandey_, Jul 23 2021
%E a(8)-a(38) from _Jon E. Schoenfield_, Jul 24 2021