

A299733


Prime numbers represented in more than one way by cyclotomic binary forms f(x,y) with x and y prime numbers and y < x.


9




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



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)
{
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!"))
)
}
(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


CROSSREFS



KEYWORD

nonn,bref,more,hard,changed


AUTHOR



STATUS

approved



