cp's OEIS Frontend

A325143 Primes represented by cyclotomic binary forms.

%I A325143 #18 May 20 2019 06:34:25
%S A325143 3,5,7,11,13,17,19,29,31,37,41,43,53,61,67,73,79,89,97,101,103,109,
%T A325143 113,127,137,139,149,151,157,163,173,181,193,197,199,211,223,229,233,
%U A325143 241,257,269,271,277,281,283,293,307,313,317,331,337,349,353,367,373
%N A325143 Primes represented by cyclotomic binary forms.
%C A325143 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.
%H A325143 Peter Luschny, <a href="/A325143/b325143.txt">Table of n, a(n) for n = 1..10000</a>
%H A325143 Étienne Fouvry, Claude Levesque, Michel Waldschmidt, <a href="http://dx.doi.org/10.4064/aa171012-24-12">Representation of integers by cyclotomic binary forms</a>, Acta Arithmetica 184 (2018), 67-86; <a href="https://arxiv.org/abs/1712.09019">arXiv:1712.09019</a>, arXiv:1712.09019 [math.NT], 2017.
%o A325143 (Julia) using Nemo
%o A325143 function isA325143(n)
%o A325143     (n < 3 || !isprime(ZZ(n))) && return false
%o A325143     R, x = PolynomialRing(ZZ, "x")
%o A325143     K = floor(Int, 5.383*log(n)^1.161) # Bounds from
%o A325143     M = floor(Int, 2*sqrt(n/3)) # Fouvry & Levesque & Waldschmidt
%o A325143     N = QQ(n)
%o A325143     for k in 3:K
%o A325143         e = Int(eulerphi(ZZ(k)))
%o A325143         c = cyclotomic(k, x)
%o A325143         for m in 1:M, j in 0:M if max(j, m) > 1
%o A325143             N == m^e*subst(c, QQ(j,m)) && return true
%o A325143     end end end
%o A325143     return false
%o A325143 end
%o A325143 [n for n in 1:373 if isA325143(n)] |> println
%Y A325143 Subsequence of A296095. Complement A325145. Number of A325141.
%Y A325143 Cf. A293654, A299214, A299498, A299733, A299928, A299930, A299956, A299964.
%K A325143 nonn
%O A325143 1,1
%A A325143 _Peter Luschny_, May 16 2019