%I #16 Apr 06 2024 14:59:29
%S 19,37,79,97,109,127,139,163,223,229,277,283,313,349,397,421,433,439,
%T 457,607,643,691,727,733,739,877,937,997,1063,1093,1327,1423,1459,
%U 1489,1567,1579,1597,1627,1657,1699,1753,1777,1801,1987,1999,2017,2089,2113,2203
%N Prime numbers represented by a cyclotomic binary form f(x, y) with x and y odd prime numbers and x > y.
%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 We say a prime number p decomposes into x and y if x and y are odd prime numbers and there exists a cyclotomic binary form f such that p = f(x,y). The transitive closure of this relation can be displayed as a binary tree, the cbf-tree of p. A cbf-tree is squarefree if all its leafs are distinct. Examples are:
%C .
%C 33751 23833 310567
%C / \ / \ / \
%C 131 79 163 19 359 283
%C / \ / \ / \ / \
%C 7 3 11 3 5 3 19 13
%C / \
%C 5 3
%C .
%C The leaves of these trees are in A299956. Related to the question whether the root of a cbf-tree can be reconstructed from its leafs is A299733.
%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.
%o (Julia)
%o using Nemo
%o function isA299930(n)
%o !isprime(ZZ(n)) && return false
%o R, z = PolynomialRing(ZZ, "z")
%o K = Int(floor(5.383*log(n)^1.161)) # Bounds from
%o M = Int(floor(2*sqrt(n/3))) # Fouvry & Levesque & Waldschmidt
%o N = QQ(n)
%o P(u) = (p for p in u:M if isprime(ZZ(p)))
%o for k in 3:K
%o e = Int(eulerphi(ZZ(k)))
%o c = cyclotomic(k, z)
%o for y in P(3), x in P(y+2)
%o N == y^e*subst(c, QQ(x, y)) && return true
%o end end
%o return false
%o end
%o A299930list(upto) = [n for n in 1:upto if isA299930(n)]
%o println(A299930list(2203))
%Y Cf. A299956 (complement), A293654, A296095, A299214, A299498, A299733, A299928, A299929, A299956, A299964.
%K nonn
%O 1,1
%A _Peter Luschny_, Feb 25 2018