> Is a decimal 23-digit numbers 11111111111111111111111 prime ?
> Could you tell me the answer with proof? >>
>
> Giving proof of primality in a text message? Hrm. Interesting.
Ok, here's a proof by converse-of-Fermat:
Let p be the number above. Then p-1 has the prime factorization:
2 * 5 * 11 * 11 * 23 * 4093 * 8779 * 21649 * 513239
All we need to do is find a value "a" such that a^((p-1)/ q) is not
equivalent to 1 modulo p for all of the primes q which divide p - 1.
(Actually, this is stricter than necessary, as a different "a" could be
chosen for each q, but never mind.)
I claim that a=11 solves this problem. The values of 11^((p-1)/q) for each
value of q is:
2 11111111111111111111110
5 5377703061176866466164
11 9819808773394829497336
23 1000
4093 6869680499138125330855
8779 5523680250213453961701
21649 8541468742226406455944
513239 10285654293302278381846
Q.E.D.
To do these computations, I used the GMP calculator at
http://www.swox.com/gmp/index.html
Paul
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