

A231613


Numbers n such that the four sixthdegree cyclotomic polynomials are simultaneously prime.


4



32034, 162006, 339105, 458811, 1780425, 2989119, 2993100, 3080205, 4375404, 6129597, 6280221, 7565142, 8489820, 10268277, 11343741, 12065076, 13067295, 13333182, 15866508, 16472802, 17040537, 18028605, 19066758, 22633629, 24256362, 24365259, 25031349
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OFFSET

1,1


COMMENTS

The polynomials are cyclotomic(7,x) = 1 + x + x^2 + x^3 + x^4 + x^5 + x^6, cyclotomic(9,x) = 1 + x^3 + x^6, cyclotomic(14,x) = 1  x + x^2  x^3 + x^4  x^5 + x^6, and cyclotomic(18,x) = 1  x^3 + x^6. The numbers 7, 9, 14 and 18 are in the sixth 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] < 30, n++; If[PrimeQ[Cyclotomic[7, n]] && PrimeQ[Cyclotomic[9, n]] && PrimeQ[Cyclotomic[14, n]] && PrimeQ[Cyclotomic[18, n]], AppendTo[t, n]]]; t
Select[Range[251*10^5], AllTrue[Cyclotomic[{7, 9, 14, 18}, #], PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 29 2016 *)


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. A231614 (similar, but with eighthdegree cyclotomic polynomials).


KEYWORD

nonn


AUTHOR



STATUS

approved



