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A299928 Integers represented by a cyclotomic binary form f(x, y) where x and y are prime numbers and 0 < y < x. 8
7, 13, 19, 29, 34, 37, 39, 49, 53, 55, 58, 61, 67, 74, 79, 91, 93, 97, 103, 109, 125, 127, 129, 130, 139, 146, 147, 163, 170, 173, 178, 194, 199, 201, 211, 217, 218, 219, 223, 229, 237, 247, 259, 273, 277, 283, 290, 291, 293, 298, 309, 313, 314, 327, 338, 349 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

REFERENCES

Trygve Nagell, Sur les représentations de l’unité par les formes binaires biquadratiques du premier rang, Arkiv för Mat. 5 (6), (1965), 477-521, (p. 517).

LINKS

Table of n, a(n) for n=1..56.

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

EXAMPLE

There are exactly four ways to represent 13 by a cyclotomic binary form f(x,y) if we require x > y > 0. In one case, x and y are prime.

13 = f(2, 1) where f(x, y) = x^4 - x^2*y^2 + y^4,

13 = f(3, 1) where f(x, y) = x^2 + x*y + y^2,

13 = f(3, 2) where f(x, y) = x^2 + y^2,

13 = f(4, 3) where f(x, y) = x^2 - x*y + y^2.

PROG

(Julia)

using Nemo

function isA299928(n)

    R, z = PolynomialRing(ZZ, "z")

    K = Int(floor(5.383*log(n)^1.161)) # Bounds from

    M = Int(floor(2*sqrt(n/3)))  # Fouvry & Levesque & Waldschmidt

    N = QQ(n)

    P(u) = (p for p in u:M if isprime(ZZ(p)))

    for k in 3:K

        e = Int(eulerphi(ZZ(k)))

        c = cyclotomic(k, z)

        for y in P(2), x in P(y+1)

            N == y^e*subst(c, QQ(x, y)) && return true

        end

    end

    return false

end

A299928list(upto) = [n for n in 1:upto if isA299928(n)]

println(A299928list(350))

CROSSREFS

Cf. A299929 (represented primes), A293654, A296095, A299214, A299498, A299733, A299930, A299956, A299964.

Sequence in context: A331830 A221027 A211431 * A096452 A258038 A059647

Adjacent sequences:  A299925 A299926 A299927 * A299929 A299930 A299931

KEYWORD

nonn

AUTHOR

Peter Luschny, Feb 21 2018

STATUS

approved

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Last modified September 20 17:16 EDT 2020. Contains 337265 sequences. (Running on oeis4.)