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
Étienne 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
KEYWORD
nonn
AUTHOR
Peter Luschny, Feb 21 2018
STATUS
approved