Thread: "Rare" Primes
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Old 2008-08-20, 03:03   #7
Jens K Andersen
 
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Feb 2006
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As hinted by a not entirely moronic Hungarian, it strongly depends on which "types" you allow. http://primepuzzles.net/puzzles/puzz_225.htm has some possibilities.

It's easy to construct rare prime forms by picking a quickly growing function with one or a few early primes. You mention Generalized Fermat 10^2^n+1, but there is no base b with more than 7 known primes b^2^n+1, and finding one with more than 10 looks very hard. The record is 7 for b=2072005925466 at http://primepuzzles.net/puzzles/puzz_399.htm

If you want relatively notable named forms then some candidates are at http://en.wikipedia.org/wiki/List_of_prime_numbers (look for comments like "only known").

In addition to your list of proven repunit primes, there are known probable primes for n = 49081, 86453, 109297, 270343.

There is no known Wall-Sun-Sun prime although infinitely many are expected to exist.
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