A185121 Smallest prime factor of 10^(2^n) + 1.

11, 101, 73, 17, 353, 19841, 1265011073, 257, 10753, 1514497, 1856104284667693057, 106907803649, 458924033, 3635898263938497962802538435084289

Offset: 0

Author

Sergio Pimentel, Jan 22 2012

Keywords

• nonn
• more
• hard

Comments

10^k+1 can only be prime if k is a power of 2. So far the only known primes of this form are a(0) = 11 and a(1) = 101. [Edited by M. F. Hasler, Aug 03 2019]

a(n) >= 2^(n+1)+1; we have a(n) = 2^(n+1)+1 for n=3, n=7, and n=15.

a(14) > 10^16. - Max Alekseyev, Jun 28 2013

From the Keller link a(15)-a(20) = 65537, 8257537, 175636481, 639631361, 70254593, 167772161. - Ray Chandler, Dec 27 2013

Examples

			For n=2, a(2)=73 since 10^(2^2) + 1 = 10001 = 73 * 137.
		

Links

• FactorDB, Factorizations of 10^(2^n)+1
• Makoto Kamada, Factorizations of 100...001
• Wilfrid Keller, Prime factors of generalized Fermat numbers Fm(10) and complete factoring status

Crossrefs

Cf. A038371, A000533, A000215, A093179, A102050

Essentially the same as A102050. - Sean A. Irvine, Feb 17 2013

Programs

Mathematica

Table[With[{k = 2^n}, FactorInteger[10^k + 1]][[1, 1]], {n, 0, 13, 1}] (* Vincenzo Librandi, Jul 23 2013 *)

PARI

a(n) = factor(10^(2^n)+1)[1, 1] \\ Michel Marcus, May 30 2013

Formula

a(n) = A038371(2^n). - M. F. Hasler, Jul 30 2019