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 primitively represented by f if f(x,y) = n has an integer solution such that x is prime to y.
LINKS
Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.
PROG
(Julia)
using Nemo
function isA299498(n)
isPrimeTo(n, k) = gcd(ZZ(n), ZZ(k)) == ZZ(1)
R, x = PolynomialRing(ZZ, "x")
K = Int(floor(5.383*log(n)^1.161)) # Bounds from
M = Int(floor(2*sqrt(n/3))) # Fouvry & Levesque & Waldschmidt
N = QQ(n)
for k in 3:K
e = Int(eulerphi(ZZ(k)))
c = cyclotomic(k, x)
for m in 1:M, j in m+1:M if isPrimeTo(m, j)
N == m^e*subst(c, QQ(j, m)) && return true
end end end
return false
end
A299498list(upto) = [n for n in 1:upto if isA299498(n)]
print(A299498list(170))
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Feb 25 2018
STATUS
approved