cp's OEIS Frontend

A299928 Integers represented by a cyclotomic binary form f(x, y) where x and y are prime numbers and 0 < y < x.

%I A299928 #11 Apr 06 2024 14:59:47
%S A299928 7,13,19,29,34,37,39,49,53,55,58,61,67,74,79,91,93,97,103,109,125,127,
%T A299928 129,130,139,146,147,163,170,173,178,194,199,201,211,217,218,219,223,
%U A299928 229,237,247,259,273,277,283,290,291,293,298,309,313,314,327,338,349
%N A299928 Integers represented by a cyclotomic binary form f(x, y) where x and y are prime numbers and 0 < y < x.
%C A299928 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.
%D A299928 Trygve Nagell, Sur les représentations de l’unité par les formes binaires biquadratiques du premier rang, Arkiv för Mat. 5 (6), (1965), 477-521, (p. 517).
%H A299928 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 A299928 There are exactly four ways to represent 13 by a cyclotomic binary form f(x,y) if we require x > y > 0. In one case, x and y are prime.
%e A299928 13 = f(2, 1) where f(x, y) = x^4 - x^2*y^2 + y^4,
%e A299928 13 = f(3, 1) where f(x, y) = x^2 + x*y + y^2,
%e A299928 13 = f(3, 2) where f(x, y) = x^2 + y^2,
%e A299928 13 = f(4, 3) where f(x, y) = x^2 - x*y + y^2.
%o A299928 (Julia)
%o A299928 using Nemo
%o A299928 function isA299928(n)
%o A299928     R, z = PolynomialRing(ZZ, "z")
%o A299928     K = Int(floor(5.383*log(n)^1.161)) # Bounds from
%o A299928     M = Int(floor(2*sqrt(n/3)))  # Fouvry & Levesque & Waldschmidt
%o A299928     N = QQ(n)
%o A299928     P(u) = (p for p in u:M if isprime(ZZ(p)))
%o A299928     for k in 3:K
%o A299928         e = Int(eulerphi(ZZ(k)))
%o A299928         c = cyclotomic(k, z)
%o A299928         for y in P(2), x in P(y+1)
%o A299928             N == y^e*subst(c, QQ(x, y)) && return true
%o A299928         end
%o A299928     end
%o A299928     return false
%o A299928 end
%o A299928 A299928list(upto) = [n for n in 1:upto if isA299928(n)]
%o A299928 println(A299928list(350))
%Y A299928 Cf. A299929 (represented primes), A293654, A296095, A299214, A299498, A299733, A299930, A299956, A299964.
%K A299928 nonn
%O A299928 1,1
%A A299928 _Peter Luschny_, Feb 21 2018