

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

See A087277.


LINKS

Table of n, a(n) for n=1..29.


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).
Sequence in context: A071067 A321246 A060069 * A296104 A170995 A319022
Adjacent sequences: A231609 A231610 A231611 * A231613 A231614 A231615


KEYWORD

nonn


AUTHOR

T. D. Noe, Dec 11 2013


STATUS

approved



