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%I #5 Dec 13 2013 12:34:19
%S 4069124,8919014,8942756,46503870,75151624,82805744,189326670,
%T 197155324,271490544,365746304,648120564,1031944990
%N Numbers n such that the five eighth-degree cyclotomic polynomials are simultaneously prime.
%C The polynomials are cyclotomic(15,x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6 + x^7 + x^8, cyclotomic(16,x) = 1 + x^8, cyclotomic(20,x) = 1 - x^2 + x^4 - x^6 + x^8, cyclotomic(24,x) = 1 - x^4 + x^8, and cyclotomic(30,x) = 1 + x - x^3 - x^4 - x^5 + x^7 + x^8. The numbers 15, 16, 20, 24 and 30 are in the eighth row of A032447.
%C By Schinzel's hypothesis H, there are an infinite number of n that yield simultaneous primes. Note that the two first-degree cyclotomic polynomials, x-1 and x+1, yield the twin primes for the numbers in A014574.
%D See A087277.
%t t = {}; n = 0; While[Length[t] < 6, n++; If[PrimeQ[Cyclotomic[15, n]] && PrimeQ[Cyclotomic[16, n]] && PrimeQ[Cyclotomic[20, n]] && PrimeQ[Cyclotomic[24, n]] && PrimeQ[Cyclotomic[30, n]], AppendTo[t, n]]]; t
%Y Cf. A014574 (first degree solutions: average of twin primes).
%Y Cf. A087277 (similar, but with second-degree cyclotomic polynomials).
%Y Cf. A231612 (similar, but with fourth-degree cyclotomic polynomials).
%Y Cf. A231613 (similar, but with sixth-degree cyclotomic polynomials).
%K nonn,more
%O 1,1
%A _T. D. Noe_, Dec 11 2013
%E Extended to 12 terms by _T. D. Noe_, Dec 13 2013