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Replies: 4   Last Post: Mar 9, 2009 2:57 PM

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 Osher Doctorow Ph.D. Posts: 30 From: Southern California Registered: 3/10/07
Posted: Mar 9, 2009 12:52 AM

>From Osher Doctorow

Readers can look at my previous replies on the thread on pi for some more on this topic, but here I will emphasize the "near-square primes" (see for example Wikipedia online or Wolfram/Eric Weisstein online articles on "Near-Square Primes" and similar articles). A considerable number of primes p are generated by equations of the form:

1) p = n^2 +/- m (or y^2 - x for x, y positive integers)

While (1) z = y^2 +/- x or n^2 +/- m for x, y positive integers does not guarantee by any means that z is a prime, a considerable number of primes are generated by this type of equation, with k = +/- 1, +/- 2, +/- 3, +/- 4, +/- 5. The case +/- 1 is especially interesting because with n^2 or y^2 replaced by slightly different functions, for example by 2^n, 2^k 3^j, n2^n, 2n or 2p, n!, we generate not only many primes but many important named primes like Mersenne Primes, Pierpoint Primes, Woodall primes, Sophie Germain primes, Twin Primes, Cousin Primes, Factorial Primes.

We have:

2) p = n^2 + 1 yields for various values of positive integer n the primes 2, 5, 17, 37, 101, 197, 257, 401, etc.

3) p = n^2 - 1 yields analogously to to (2) the prime 3.

4) p = n^2 + 2 yields analogously to (2) the primes 3, 11, 83, 227, 443, etc.

5) p = n^2 - 2 yields analogously to (2) the primes 2, 7, 23, 47, 79, 167, 223, 359, etc.

I'll let Readers examine the other cases mentioned (+/- 3, +/- 4, +/- 5_, except to mention that the primes 13, 61, 19, 67, 29, 53, 41 are among those generated by the latter 6.

Osher Doctorow