Is $10^{2^{21}}+1$ known to be composite?

Question

I looked at the generalized Fermat-prime-numbers. According to factordb, the case $$10^{2^{21}}+1$$ is unknown. Neither a factor is displayed nor $C$ for "composite". Hence my question :

Is $$10^{2^{21}}+1$$ be known to be composite ? Is a factor known ? If no, what is the search limit ?

If factordb says it's unknown, why do you think it is known to be composite or a factor is known? — lhf, Feb 13 '19 at 15:20
See "$10^{2^{n}}+1$ has no (known?) prime factors for $n=13$, $14$, $21$, $23$, $24$, $25$, $\ldots $" http://www.ams.org/journals/mcom/1998-67-221/S0025-5718-98-00891-6/S0025-5718-98-00891-6.pdf — Winther, Feb 13 '19 at 15:20
Possible duplicate of Is there a prime which is the form of $10^n + 1$ except $2, 11, 101$? — Winther, Feb 13 '19 at 15:20
@Winther Thank you. What remains is whether the number has been verified for primality. — Peter, Feb 13 '19 at 15:46
@Winther The answer of the linked question does not cover my question. Does your link contain a statement about the primality and/or a search limit for the prime factors ? — Peter, Feb 13 '19 at 15:51
@Ihf Because I do not assume that every verified number or every found factor is in factordb. — Peter, Feb 13 '19 at 15:55
@Ihf Moreover, the status of a number cannot be entered manually in factordb. The program itself must check it. — Peter, Feb 13 '19 at 16:04
A primality check using Maple would be to do this: N := 10^(2^21)+1: Power(2,N-1) mod N: evalb(% = 1); but based smaller examples, this might take a month. — Mark, Feb 13 '19 at 17:53

score 3 · Accepted Answer · answered Feb 16 '19 at 09:40

3

PFGW: $$10^{2^{21}}+1$$ is composite: RES64: [651AE2CA2B659AF9] (175710.6149s+0.0590s)

answered Feb 16 '19 at 09:40

Enzo Creti

1

0x651AE2CA2B659AF9 does not divide 10^(2^21)+1. – Jun 17 '19 at 00:28

Is $10^{2^{21}}+1$ known to be composite?

1 Answers1