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%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
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