Quote:
Originally Posted by Heck
Eventually we will want to find the smallest tenmilliondigit prime.
mighty big number, factor
10^9999999 + 1, 7
10^9999999 + 3, 23
10^9999999 + 7, 11852969
10^9999999 + 9, 47
10^9999999 + 11, 3
10^9999999 + 13, 277
10^9999999 + 17, 3
10^9999999 + 19, 7583
10^9999999 + 21, 718453
10^9999999 + 23, 3
10^9999999 + 27, 13
10^9999999 + 29, 3
10^9999999 + 31, 821879
etc..

This gives us the following moduloresults, if somebody wants to start sieving:
let a be 10^9999999; let b be 1, 3, 7, ...
a+1 ≡ 2 mod 3
a+1 ≡ 1 mod 5 (obviously)
a+1 ≡ 0 mod 7
a+1 ≡ 0 mod 11 (each number of the form 1000000....00001 which has an even number of digits is divisible by 11; (a+1)/11 = 90909090.......09091; 9999998 digits)
a+1 ≡ 0 mod 13
a+1 ≡ ??? mod 17
a+1 ≡ ??? mod 19
a+1 ≡ 21 mod 23
a+1 ≡ ??? mod 29, 31, 37, 41, 43
a+1 ≡ 39 mod 47
It gives us also two more factors of 10^9999999+1, namely 11 and 13.