

A231614


Numbers n such that the five eighthdegree cyclotomic polynomials are simultaneously prime.


4



4069124, 8919014, 8942756, 46503870, 75151624, 82805744, 189326670, 197155324, 271490544, 365746304, 648120564, 1031944990
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OFFSET

1,1


COMMENTS

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.
By Schinzel's hypothesis H, there are an infinite number of n that yield simultaneous primes. Note that the two firstdegree cyclotomic polynomials, x1 and x+1, yield the twin primes for the numbers in A014574.


REFERENCES



LINKS



MATHEMATICA

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


CROSSREFS

Cf. A014574 (first degree solutions: average of twin primes).
Cf. A087277 (similar, but with seconddegree cyclotomic polynomials).
Cf. A231612 (similar, but with fourthdegree cyclotomic polynomials).
Cf. A231613 (similar, but with sixthdegree cyclotomic polynomials).


KEYWORD

nonn,more


AUTHOR



EXTENSIONS

Extended to 12 terms by T. D. Noe, Dec 13 2013


STATUS

approved



