

A231612


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


4



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
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OFFSET

1,1


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 firstdegree cyclotomic polynomials, x1 and x+1, yield the twin primes for the numbers in A014574.


REFERENCES



LINKS



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 *)


CROSSREFS

Cf. A014574 (first degree solutions: average of twin primes).
Cf. A087277 (similar, but with seconddegree cyclotomic polynomials).
Cf. A231613 (similar, but with sixthdegree cyclotomic polynomials).
Cf. A231614 (similar, but with eighthdegree cyclotomic polynomials).


KEYWORD

nonn


AUTHOR



STATUS

approved



