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

1,1

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

Table of n, a(n) for n=1..3.

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

Subsequence of A299929.

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

Sequence in context: A362301 A142170 A069593 * A086120 A129701 A221746

Adjacent sequences: A299730 A299731 A299732 * A299734 A299735 A299736

Keyword

nonn,bref,more,hard

Author

Peter Luschny, Feb 21 2018

Status

approved