A153438 Least k > 1 such that k^(3^n)*(k^(3^n)+1) + 1 is prime.

2, 2, 2, 21, 209, 72, 260, 17, 3311, 4469, 94259, 55599

Offset: 0

Author

Pierre CAMI, Dec 26 2008

Keywords

• nonn
• more
• hard

Comments

Numbers of the form k^n*(k^n+1) + 1 with n > 0, k > 1 may be primes only if n has the form 3^j. When n is even k^(4*n) + k^(2*n) + 1 = (k^(2*n)+1)^2 - (k^n)^2 = (k^(2*n)+k^n+1)*(k^(2*n)-k^n+1) so composite. But why if n odd > 3 and not a power of 3 is k^n*(k^n+1) + 1 always composite?

Phi[3^(n+1),k] = k^(3^n)*(k^(3^n)+1)+1. When m <> 3^n in k^m*(k^m+1)+1, Phi[3m,k] < k^m*(k^m+1)+1 and is a divisor of it. - Lei Zhou, Feb 09 2012

The prime number corresponding to the 10th term is a 587458-digit number. - Lei Zhou, Jul 04 2014

x^(2*k) + x^k + 1 = (x^(3*k) - 1)/(x^k - 1) is the product over n dividing 3k but not dividing k of cyclotomic polynomials Phi(n). If k is a power of 3, n = 3k is the only such divisor and we have a single irreducible cyclotomic polynomial Phi(3k). Otherwise we have the product of more than one polynomial, with integer values > 1 for integer x > 1, and thus always composite numbers. - Martin Becker, Jun 22 2021

Links

• Lei Zhou, Prime Database Entry, July 04, 2014.
• R. Propper and S. Batalov, Prime Database Entry, Nov 17, 2023.

Crossrefs

Cf. A056993, A085398, A101406, A153436.

Programs

Mathematica

Table[i = 1; m = 3^u; While[i++; cp = 1 + i^m + i^(2*m); ! PrimeQ[cp]]; i, {u, 1, 7}] (* Lei Zhou, Feb 01 2012 *)

PARI

a(n) = my(k=2); while (!isprime(k^(3^n)*(k^(3^n)+1) + 1), k++); k; \\ Michel Marcus, Jan 01 2023

Formula

a(n) = A085398(3^(n+1)). - Jinyuan Wang, Jan 01 2023

Extensions

3311 from Lei Zhou using OpenPFGW, Feb 01 2012

4469 from Lei Zhou using OpenPFGW, Feb 09 2012

New term, 94259, from Lei Zhou using OpenPFGW, Jul 04 2014

Name and Comment corrected by Robert Price, Nov 11 2018

a(0) inserted by Jinyuan Wang, Jan 01 2023

a(11) from Ryan Propper and Serge Batalov, Nov 17 2023