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A153438 Least k > 1 such that k^(3^n)*(k^(3^n)+1) + 1 is prime. 13
2, 2, 2, 21, 209, 72, 260, 17, 3311, 4469, 94259, 55599 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,1
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.
FORMULA
a(n) = A085398(3^(n+1)). - Jinyuan Wang, Jan 01 2023
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 *)
PROG
(PARI) a(n) = my(k=2); while (!isprime(k^(3^n)*(k^(3^n)+1) + 1), k++); k; \\ Michel Marcus, Jan 01 2023
CROSSREFS
Sequence in context: A140283 A067097 A260082 * A098915 A095855 A157979
KEYWORD
nonn,more,hard
AUTHOR
Pierre CAMI, Dec 26 2008
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
STATUS
approved

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Last modified June 19 06:59 EDT 2024. Contains 373492 sequences. (Running on oeis4.)