A231612 Numbers n such that the four fourth-degree cyclotomic polynomials are simultaneously prime.

2, 90750, 194468, 229592, 388332, 868592, 1054868, 1148390, 1380380, 1415920, 1461372, 1496010, 1614800, 1706398, 1992210, 2439042, 2478212, 2644498, 2791910, 3073300, 3264448, 3824370, 3892780, 3939222, 3941938, 4425970, 4468980, 4594138, 4683700

Offset: 1

Author

T. D. Noe, Dec 11 2013

Keywords

• nonn

Comments

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.

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.

References

• See A087277.

Crossrefs

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

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

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

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

Programs

Mathematica

Select[Range[5000000], PrimeQ[Cyclotomic[5, #]] && PrimeQ[Cyclotomic[8, #]] && PrimeQ[Cyclotomic[10, #]] && PrimeQ[Cyclotomic[12, #]] &]
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 *)