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A299498
Integers primitively represented by cyclotomic binary forms.
8
3, 5, 7, 10, 11, 13, 17, 19, 21, 25, 26, 29, 31, 34, 37, 39, 41, 43, 49, 50, 53, 55, 57, 58, 61, 65, 67, 73, 74, 79, 82, 85, 89, 91, 93, 97, 101, 103, 106, 109, 111, 113, 121, 122, 125, 127, 129, 130, 133, 137, 139, 145, 146, 147, 149, 151, 157, 163, 169, 170
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))
KEYWORD
nonn
AUTHOR
Peter Luschny, Feb 25 2018
STATUS
approved