A231612 - OEIS

%I #5 Feb 22 2018 18:43:27

%S 2,90750,194468,229592,388332,868592,1054868,1148390,1380380,1415920,

%T 1461372,1496010,1614800,1706398,1992210,2439042,2478212,2644498,

%U 2791910,3073300,3264448,3824370,3892780,3939222,3941938,4425970,4468980,4594138,4683700

%N Numbers n such that the four fourth-degree cyclotomic polynomials are simultaneously prime.

%C The polynomials are cyclotomic(5,x) = 1 + x + x^2 + x^3 + x^4, cyclotomic(8,x) = 1 + x^4, cyclotomic(10,x) = 1 - x + x^2 - x^3 + x^4, and cyclotomic(12,x) = 1 - x^2 + x^4. The numbers 5, 8, 10, and 12 are in the fourth 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 Select[Range[5000000], PrimeQ[Cyclotomic[5, #]] && PrimeQ[Cyclotomic[8, #]] && PrimeQ[Cyclotomic[10, #]] && PrimeQ[Cyclotomic[12, #]] &]

%t Select[Range[47*10^5],AllTrue[Thread[Cyclotomic[{5,8,10,12},#]],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Feb 22 2018 *)

%Y Cf. A014574 (first degree solutions: average of twin primes).

%Y Cf. A087277 (similar, but with second-degree cyclotomic polynomials).

%Y Cf. A231613 (similar, but with sixth-degree cyclotomic polynomials).

%Y Cf. A231614 (similar, but with eighth-degree cyclotomic polynomials).

%K nonn

%O 1,1

%A _T. D. Noe_, Dec 11 2013