Thread: "Rare" Primes
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Old 2008-09-07, 17:45   #60
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Aug 2006

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Originally Posted by Housemouse View Post
On the site Rudolf Ondrejka lists ten rare primes. One example he refers to as a beastly palindrome of the type (10^n 666)*10n-2+1.

Has it been proven that only 7 exist?
No. Given that (10^n + 666) * 10^(n-2) + 1 is not divisible by 2, 3, or 5, a quick guess at the 'chance' it's prime as
15/4 * (1/log(10^(2n-2)))

The sum of this from 2 to 3000 is 6.98, so having 7 from n = 2 to 3000 is pretty much what you'd expect. The expected number up to a million is 11.72, so it would be unusual if only 7 existed. In fact, since the harmonic series diverges, you'd naively expect an infinite number of such primes.

Originally Posted by Housemouse View Post
Wikipedia, states that there is only one positive Genocchi prime; has this been proven?
MathWorld has "D. Terr (pers. comm., Jun. 8, 2004) proved that these are in fact, the only prime Genocchi numbers.".
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