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2004-10-25, 13:15   #4
Andi47

Oct 2004
Austria

2×17×73 Posts

Quote:
 Originally Posted by Heck Eventually we will want to find the smallest ten-million-digit 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 modulo-results, 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.