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A299929 Prime numbers represented by a cyclotomic binary form f(x, y) with x and y prime numbers and 0 < y < x. 6


%S 7,13,19,29,37,53,61,67,79,97,103,109,127,139,163,173,199,211,223,229,

%T 277,283,293,313,349,397,421,433,439,457,463,487,541,577,607,641,643,

%U 691,727,733,739,787,877,937,997,1009,1031,1063,1093,1327,1373,1423,1447

%N Prime numbers represented by a cyclotomic binary form f(x, y) with x and y prime numbers and 0 < y < x.

%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.

%H Etienne 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 6841 = f(7,5) for f(x,y) = x^4+x^3*y+x^2*y^2+x*y^3+y^4.

%t isA299929[n_] := If[! PrimeQ[n], Return[False],

%t K = Floor[5.383 Log[n]^1.161]; M = Floor[2 Sqrt[n/3]];

%t For[k = 3, k <= K, k++,

%t For[y = 1, y <= M, y++, If[PrimeQ[y], For[x = y + 1, x <= M, x++, If[PrimeQ[x],

%t If[n == y^EulerPhi[k] Cyclotomic[k, x/y], Return[True]]]]]]];

%t Return[False]]; Select[Range[1450], isA299929]

%o (Julia)

%o A299929list(upto) = [n for n in 1:upto if isprime(ZZ(n)) && isA299928(n)]

%o println(A299929list(1450))

%Y Cf. A293654, A296095, A299214, A299498, A299733, A299928, A299930, A299956, A299964.

%K nonn

%O 1,1

%A _Peter Luschny_, Feb 21 2018

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Last modified September 18 04:32 EDT 2020. Contains 337165 sequences. (Running on oeis4.)