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

7, 13, 19, 29, 37, 53, 61, 67, 79, 97, 103, 109, 127, 139, 163, 173, 199, 211, 223, 229, 277, 283, 293, 313, 349, 397, 421, 433, 439, 457, 463, 487, 541, 577, 607, 641, 643, 691, 727, 733, 739, 787, 877, 937, 997, 1009, 1031, 1063, 1093, 1327, 1373, 1423, 1447

Offset: 1

Author

Peter Luschny, Feb 21 2018

Keywords

• nonn

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.

Examples

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

Links

• Étienne Fouvry, Claude Levesque, Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.

Crossrefs

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

Programs

Julia

A299929list(upto) = [n for n in 1:upto if isprime(ZZ(n)) && isA299928(n)]
println(A299929list(1450))

Mathematica

isA299929[n_] := If[! PrimeQ[n], Return[False],
   K = Floor[5.383 Log[n]^1.161]; M = Floor[2 Sqrt[n/3]];
   For[k = 3, k <= K, k++,
   For[y = 1, y <= M, y++, If[PrimeQ[y], For[x = y + 1, x <= M, x++, If[PrimeQ[x],
   If[n == y^EulerPhi[k] Cyclotomic[k, x/y], Return[True]]]]]]];
Return[False]]; Select[Range[1450], isA299929]