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Old 2018-07-18, 03:57   #2
GP2
 
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Sep 2003

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Lack of small factors doesn't mean much. After all, for Mersenne numbers in the typical ranges we work on, we find small factors for only about 60% of them. For the rest we have to do Lucas-Lehmer tests, but nearly all of them test composite and Mersenne primes are exceedingly rare.

For Fermat numbers the counterpart to the Lucas-Lehmer test is the Pépin test, but F33 has an exponent of about 8.6 billion so it's just too big to do this test on the current generation of computers.

I'd imagine that the odds of finding a Fermat prime decrease as the exponent increases, similar to what is observed for Mersenne numbers. For Mersennes we have the Wagstaff conjecture which matches the existing data pretty well. At each order of magnitude the number of Mersenne primes is roughly the same: we expect to find as many primes with exponent between 1,000 and 10,000 as the number with exponent between 1 million and 10 million or between 1 billion and 10 billion.

At some point in the future it may become practical to do ECM testing on F33, but not yet. The largest Fermat number for which ECM testing is practicable is F29, with an exponent of about 537 million, which already has a known small factor = 1120049 × 229+2 + 1. I have been doing some ECM testing on it to try to find a second factor, but it takes about 120GB of memory and about a week to do a single ECM curve for F29.

Last fiddled with by GP2 on 2018-07-18 at 04:26
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