Yesterday I figured I’ll try to factor the first few numbers greater than 10¹⁰⁰⁰⁰⁰⁰⁰⁰ (or at least find their smallest divisors). The idea was to see how hard it is to find the divisors of the first few hundred 100,000,001-digit numbers.
I took a pretty simple approach: for each prime number p, calculate 10¹⁰⁰⁰⁰⁰⁰⁰⁰%p (the remainder after dividing by p). Then I used the result to create a partial sieve. After the first 300,000 primes, I still couldn’t find a divisor for 10¹⁰⁰⁰⁰⁰⁰⁰⁰+37.
Update
10¹⁰⁰⁰⁰⁰⁰⁰⁰+37 is divisible by 6,870,527 (the 468,407th prime). The next one without a known divisor is 10⁹⁹⁹⁹⁹⁹⁹⁹+69 (tried to divide by the first 2,348,559 primes, no factors found so far).
Here’s the smallest prime divisor for the first few 100,000,001-digit integers
(I = 10¹⁰⁰⁰⁰⁰⁰⁰⁰):
.------------------.
| number | divisor |
---------+----------
| I | 2 |
| 1 + I | 10753 |
| 2 + I | 2 |
| 3 + I | 7 |
| 4 + I | 2 |
| 5 + I | 3 |
| 6 + I | 2 |
| 7 + I | 113 |
| 8 + I | 2 |
| 9 + I | 197 |
| 10 + I | 2 |
| 11 + I | 3 |
| 12 + I | 2 |
| 13 + I | 29 |
| 14 + I | 2 |
| 15 + I | 5 |
| 16 + I | 2 |
| 17 + I | 3 |
| 18 + I | 2 |
| 19 + I | 2087 |
| 20 + I | 2 |
| 21 + I | 11 |
| 22 + I | 2 |
| 23 + I | 3 |
| 24 + I | 2 |
| 25 + I | 5 |
| 26 + I | 2 |
| 27 + I | 37 |
| 28 + I | 2 |
| 29 + I | 3 |
| 30 + I | 2 |
| 31 + I | 7 |
| 32 + I | 2 |
| 33 + I | 17 |
| 34 + I | 2 |
| 35 + I | 3 |
| 36 + I | 2 |
| 37 + I | 6870527 |
| 38 + I | 2 |
| 39 + I | 139 |
| 40 + I | 2 |
| 41 + I | 3 |
| 42 + I | 2 |
| 43 + I | 11 |
| 44 + I | 2 |
| 45 + I | 5 |
| 46 + I | 2 |
| 47 + I | 3 |
| 48 + I | 2 |
| 49 + I | 13 |
| 50 + I | 2 |
| 51 + I | 71 |
| 52 + I | 2 |
| 53 + I | 3 |
| 54 + I | 2 |
| 55 + I | 5 |
| 56 + I | 2 |
| 57 + I | 31 |
| 58 + I | 2 |
| 59 + I | 3 |
| 60 + I | 2 |
| 61 + I | 59 |
| 62 + I | 2 |
| 63 + I | 399389 |
| 64 + I | 2 |
| 65 + I | 3 |
| 66 + I | 2 |
| 67 + I | 17 |
| 68 + I | 2 |
'------------------'