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.
We say a prime number p decomposes into x and y if x and y are odd prime numbers and there exists a cyclotomic binary form f such that p = f(x,y). The transitive closure of this relation can be displayed as a binary tree, the cbf-tree of p. A cbf-tree is squarefree if all its leafs are distinct. Examples are:
.
33751 23833 310567
/ \ / \ / \
131 79 163 19 359 283
/ \ / \ / \ / \
7 3 11 3 5 3 19 13
/ \
5 3
.
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 isA299930(n)
!isprime(ZZ(n)) && return false
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(3), x in P(y+2)
N == y^e*subst(c, QQ(x, y)) && return true
end end
return false
end
A299930list(upto) = [n for n in 1:upto if isA299930(n)]
println(A299930list(2203))
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Feb 25 2018
STATUS
approved