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 A299930 Prime numbers represented by a cyclotomic binary form f(x, y) with x and y odd prime numbers and x > y. 9
 19, 37, 79, 97, 109, 127, 139, 163, 223, 229, 277, 283, 313, 349, 397, 421, 433, 439, 457, 607, 643, 691, 727, 733, 739, 877, 937, 997, 1063, 1093, 1327, 1423, 1459, 1489, 1567, 1579, 1597, 1627, 1657, 1699, 1753, 1777, 1801, 1987, 1999, 2017, 2089, 2113, 2203 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. 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: .       33751                   23833                   310567        / \                     /  \                    /  \     131   79                163    19               359    283           / \               / \    / \                     / \          7   3            11   3  5   3                  19   13                                                         /  \                                                        5    3 . The leafs 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. LINKS Etienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017. PROG (Julia) using Nemo function isA299930(n)     !isprime(ZZ(n)) && return false     R, z = PolynomialRing(ZZ, "z")     K = Int(floor(5.383*log(n)^1.161)) # Bounds from     M = Int(floor(2*sqrt(n/3)))  # Fouvry & Levesque & Waldschmidt     N = QQ(n)     P(u) = (p for p in u:M if isprime(ZZ(p)))     for k in 3:K         e = Int(eulerphi(ZZ(k)))         c = cyclotomic(k, z)         for y in P(3), x in P(y+2)             N == y^e*subst(c, QQ(x, y)) && return true     end end     return false end A299930list(upto) = [n for n in 1:upto if isA299930(n)] println(A299930list(2203)) CROSSREFS Cf. A299956 (complement), A293654, A296095, A299214, A299498, A299733, A299928, A299929, A299956, A299964. Sequence in context: A245381 A050528 A257074 * A053685 A244111 A136063 Adjacent sequences:  A299927 A299928 A299929 * A299931 A299932 A299933 KEYWORD nonn AUTHOR Peter Luschny, Feb 25 2018 STATUS approved

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Last modified September 28 12:47 EDT 2022. Contains 357070 sequences. (Running on oeis4.)