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

1, 16, 61, 81, 256, 625, 976, 1296, 2401, 4096, 4941, 6561, 10000, 14641, 15616, 20736, 28561, 38125, 38416, 50625, 65536, 79056, 83521, 104976, 130321, 146461, 160000, 194041, 194481, 229981, 234256, 249856, 279841, 331776, 390625, 400221, 456976, 531441

Offset

1,2

Comments

Positive integers C such that Phi_5(x,y) = Phi_12(u,v) = C has a solution with nonzero (x,y,u,v).

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.

Links

Table of n, a(n) for n=1..38.

Étienne Fouvry, Claude Levesque and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Example

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.

Crossrefs

Cf. A296095.

Sequence in context: A296958 A258731 A316125 * A039406 A031190 A043229

Adjacent sequences: A345891 A345892 A345893 * A345895 A345896 A345897

Keyword

nonn

Author

Shashi Kant Pandey, Jul 23 2021

Extensions

a(8)-a(38) from Jon E. Schoenfield, Jul 24 2021

Status

approved