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 A345894 Positive integers representable by the two cyclotomic binary forms Phi_5(x,y) and Phi_12(u,v). 0

%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

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Last modified August 10 21:39 EDT 2024. Contains 375058 sequences. (Running on oeis4.)