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.
There are only three prime numbers below 600000 which satisfy the given conditions. No prime number below 600000 exists which has more than one representation if we require a representation by odd prime numbers y < x.
LINKS
Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.
EXAMPLE
33751 = f(131,79) for f(x,y) = x^2 + x*y + y^2.
33751 = f( 13, 2) for f(x,y) = x^4+x^3*y+x^2*y^2+x*y^3+y^4.
PROG
(PARI)
A299733(upto) =
{
my(K, M, phi, multi);
forprime(n = 2, upto, multi = 0;
K = floor(5.383*log(n)^1.161);
M = floor(2*sqrt(n/3));
for(k = 3, K,
phi = eulerphi(k);
forprime(y = 2, M,
forprime(x = y + 1, M,
if(n == y^phi*polcyclo(k, x/y),
multi += 1
)
)
)
);
if(multi > 1, print(n, " has multiple reps!"))
)
}
A299733(100000)
(Julia) using Nemo
function isA299733(n)
if n < 3 || !isprime(ZZ(n)) return false end
R, x = PolynomialRing(ZZ, "x")
K = floor(Int, 5.383*log(n)^1.161) # Bounds from
M = floor(Int, 2*sqrt(n/3)) # Fouvry & Levesque & Waldschmidt
N = QQ(n); multi = 0
for k in 3:K
e = Int(eulerphi(ZZ(k)))
c = cyclotomic(k, x)
for m in 2:M if isprime(ZZ(m))
for j in m:M if isprime(ZZ(j))
if N == m^e*subst(c, QQ(j, m)) multi += 1
end end end end end end
multi > 1
end # Peter Luschny, May 16 2019
CROSSREFS
KEYWORD
nonn,bref,more,hard
AUTHOR
Peter Luschny, Feb 21 2018
STATUS
approved