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.

19, 97, 33751

Offset: 1

Author

Peter Luschny, Feb 21 2018

Keywords

• nonn
• bref
• more
• hard

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.

Examples

			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.
		

Links

• Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Crossrefs

Subsequence of A299929.

Cf. A293654, A296095, A299214, A299498, A299928, A299930, A299956, A299964.

Programs

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

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)