Historically, John Cosgrave very succesfully searched for factors of Fermat numbers (a.k.a. Fermat divisors) by searching for primes of the form 3*2^n+1. These have a very high probability (1/3 basically) of dividing F(n-e) for some small e. (That's the n-e-th Fermat number 2^2^(n-e)+1.)
Here are the primes he found (rankings as of 2007-03-22)
| description | digits | rank | who | year |
|---|---|---|---|---|
| comment | ||||
| 3*2^2478785+1 | 746190 | 20 | g245 | 2003 |
| Divides Fermat F(2478782), GF(2478782,3), GF(2478776,6), GF(2478782,12) | ||||
| 3*2^2145353+1 | 645817 | 30 | g245 | 2003 |
| Divides Fermat F(2145351), GF(2145351,3), GF(2145352,5), GF(2145348,6), GF(2145352,10), GF(2145351,12) | ||||
| 3*2^916773+1 | 275977 | 174 | g245 | 2001 |
| Divides GF(916771,3), GF(916772,10) | ||||
| 3*2^382449+1 | 115130 | 1700 | g132 | 1999 |
| Divides Fermat F(382447), GF(382447,3), GF(382447,12), GF(382443,6) | ||||
| 3*2^362765+1 | 109204 | 2029 | g245 | 2002 |
| Divides GF(362763,12), GF(362764,10) | ||||
John has generously made his deeply pre-sieved files available, such that others can pick up where he left off. Note that the remaining candidates are very large, and it may take several days to test each one.
Another hastily constructed page by Phil Carmody
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3*2^n+1 Fermat Factor Search