A299733 - OEIS

%I #27 Apr 06 2024 15:00:04

%S 19,97,33751

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

%C 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.

%C 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.

%H Étienne Fouvry, Claude Levesque, Michel Waldschmidt, <a href="https://arxiv.org/abs/1712.09019">Representation of integers by cyclotomic binary forms</a>, arXiv:1712.09019 [math.NT], 2017.

%e 33751 = f(131,79) for f(x,y) = x^2 + x*y + y^2.

%e 33751 = f( 13, 2) for f(x,y) = x^4+x^3*y+x^2*y^2+x*y^3+y^4.

%o (PARI)

%o A299733(upto) =

%o {

%o     my(K, M, phi, multi);

%o     forprime(n = 2, upto, multi = 0;

%o         K = floor(5.383*log(n)^1.161);

%o         M = floor(2*sqrt(n/3));

%o         for(k = 3, K,

%o             phi = eulerphi(k);

%o             forprime(y = 2, M,

%o                 forprime(x = y + 1, M,

%o                     if(n == y^phi*polcyclo(k, x/y),

%o                         multi += 1

%o                     )

%o                 )

%o             )

%o         );

%o         if(multi > 1, print(n," has multiple reps!"))

%o     )

%o }

%o A299733(100000)

%o (Julia) using Nemo

%o function isA299733(n)

%o     if n < 3 || !isprime(ZZ(n)) return false end

%o     R, x = PolynomialRing(ZZ, "x")

%o     K = floor(Int, 5.383*log(n)^1.161) # Bounds from

%o     M = floor(Int, 2*sqrt(n/3)) # Fouvry & Levesque & Waldschmidt

%o     N = QQ(n); multi = 0

%o     for k in 3:K

%o         e = Int(eulerphi(ZZ(k)))

%o         c = cyclotomic(k, x)

%o         for m in 2:M if isprime(ZZ(m))

%o             for j in m:M if isprime(ZZ(j))

%o                 if N == m^e*subst(c, QQ(j,m)) multi += 1

%o     end end end end end end

%o     multi > 1

%o end # _Peter Luschny_, May 16 2019

%Y Subsequence of A299929.

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

%K nonn,bref,more,hard

%O 1,1

%A _Peter Luschny_, Feb 21 2018