A070519 Numbers k such that Cyclotomic(k,k) (i.e., the value of k-th cyclotomic polynomial at k) is a prime number.

2, 3, 4, 6, 10, 12, 14, 19, 31, 46, 74, 75, 98, 102, 126, 180, 236, 310, 368, 1770, 1858, 3512, 4878, 5730, 7547, 7990, 8636, 9378, 11262

Offset: 1

Author

Labos Elemer, May 02 2002

Keywords

• nonn
• more

Comments

When n is prime, then the solutions are given in A088790.

No term of this sequence is congruent to 1 mod 4. In general, if k = s^2*t where t is squarefree and t == 1 (mod 4), then Cyclotomic(k,t*x^2) is the product of two polynomials. See the Wikipedia link below. - Jianing Song, Sep 25 2019

All terms <= 1858 have been proven with PARI's implementation of ECPP. All larger terms are BPSW PRPs. There are no further terms <= 30000. - Lucas A. Brown, Dec 28 2020

Links

• Eric Weisstein's World of Mathematics, Cyclotomic Polynomial
• Wikipedia, Aurifeuillean factorization

Crossrefs

Cf. A070518, A070520, A088790 ((k^k-1)/(k-1) is prime), A088817 (cyclotomic(2k,k) is prime), A088875 (cyclotomic(k,-k) is prime).

Cf. A117544, A085398.

Programs

Mathematica

Do[s=Cyclotomic[n, n]; If[PrimeQ[s], Print[n]], {n, 2, 256}]

PARI

for(n=2,10^9,if(ispseudoprime(polcyclo(n,n)),print1(n,", "))); \\ Joerg Arndt, Jan 22 2015

Extensions

More terms from T. D. Noe, Oct 17 2003

a(29) from Charles R Greathouse IV, May 05 2011