cp's OEIS Frontend

A299930 Prime numbers represented by a cyclotomic binary form f(x, y) with x and y odd prime numbers and x > y.

%I A299930 #16 Apr 06 2024 14:59:29
%S A299930 19,37,79,97,109,127,139,163,223,229,277,283,313,349,397,421,433,439,
%T A299930 457,607,643,691,727,733,739,877,937,997,1063,1093,1327,1423,1459,
%U A299930 1489,1567,1579,1597,1627,1657,1699,1753,1777,1801,1987,1999,2017,2089,2113,2203
%N A299930 Prime numbers represented by a cyclotomic binary form f(x, y) with x and y odd prime numbers and x > y.
%C A299930 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 A299930 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 A299930 .
%C A299930       33751                   23833                   310567
%C A299930        / \                     /  \                    /  \
%C A299930     131   79                163    19               359    283
%C A299930           / \               / \    / \                     / \
%C A299930          7   3            11   3  5   3                  19   13
%C A299930                                                         /  \
%C A299930                                                        5    3
%C A299930 .
%C A299930 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 A299930 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 A299930 (Julia)
%o A299930 using Nemo
%o A299930 function isA299930(n)
%o A299930     !isprime(ZZ(n)) && return false
%o A299930     R, z = PolynomialRing(ZZ, "z")
%o A299930     K = Int(floor(5.383*log(n)^1.161)) # Bounds from
%o A299930     M = Int(floor(2*sqrt(n/3)))  # Fouvry & Levesque & Waldschmidt
%o A299930     N = QQ(n)
%o A299930     P(u) = (p for p in u:M if isprime(ZZ(p)))
%o A299930     for k in 3:K
%o A299930         e = Int(eulerphi(ZZ(k)))
%o A299930         c = cyclotomic(k, z)
%o A299930         for y in P(3), x in P(y+2)
%o A299930             N == y^e*subst(c, QQ(x, y)) && return true
%o A299930     end end
%o A299930     return false
%o A299930 end
%o A299930 A299930list(upto) = [n for n in 1:upto if isA299930(n)]
%o A299930 println(A299930list(2203))
%Y A299930 Cf. A299956 (complement), A293654, A296095, A299214, A299498, A299733, A299928, A299929, A299956, A299964.
%K A299930 nonn
%O A299930 1,1
%A A299930 _Peter Luschny_, Feb 25 2018