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A231612 Numbers n such that the four fourth-degree 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 (list; graph; refs; listen; history; text; internal format)
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 first-degree cyclotomic polynomials, x-1 and x+1, yield the twin primes for the numbers in A014574.
REFERENCES
See A087277.
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 second-degree cyclotomic polynomials).
Cf. A231613 (similar, but with sixth-degree cyclotomic polynomials).
Cf. A231614 (similar, but with eighth-degree cyclotomic polynomials).
Sequence in context: A371645 A060069 A354177 * A296104 A170995 A319022
KEYWORD
nonn
AUTHOR
T. D. Noe, Dec 11 2013
STATUS
approved

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Last modified May 19 00:35 EDT 2024. Contains 372666 sequences. (Running on oeis4.)