Talk:Strong prime

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Just in case anyone is wondering, 439351292910452432574786963588089477522344331 is in the ballpark of 439351292910452432574786963588089477522344721, the 138th prime of the Hoffman inversion of Wilf's primefree sequence. PrimeFan 23:12, 27 January 2006 (UTC)

The article says: "Given a twin prime {p, p + 2}, the lesser prime of the two, p, will almost certainly be a strong prime. In all the twin primes in the first ten million primes, the only lesser of a pair that is not a strong prime is 3 (from the pair {3, 5}, the arithmetic mean of 2 and 5 is 3.5)." In fact, {5, 7} is a pair of twin primes, and 5 is not strong according to the definition here (it's balanced, since 5 = (3+7)/2). In fact, 3 and 5 are the only lesser members of a pair of twin primes which are not strong primes. This is easy to show: if p and p+2 are primes, then p is not a strong prime if and only if one of p-1 or p-2 is prime. But say p >= 7. Then p is congruent to 1 or 5 mod 6. Since p+2 is prime, we must have p congruent to 5 mod 6. So neither p-1 nor p-2 is prime, since these are 4 and 3 mod 6, respectively. This proves that if p >= 7 is prime, and p+2 is prime, then p is strong. We can check p = 3, 5 "by hand", and we see that these aren't strong. I've edited the article accordingly. Izzycat 21:06, 16 February 2006 (UTC) ǖ

Thank you very much the correction and explanation. PrimeFan 22:02, 17 February 2006 (UTC)