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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 1:54 AM

>From Osher Doctorow

Some of the most interesting primes p are generated by:

1) p = 6n +/- 1, n positive integer, p prime

although 6n +/- 1 can also generate non-primes. The Twin Primes, which are Prime Pairs of form (p, p+2) are all generated by (1) except for te pair (3, 5):

2) Twin primes = (3, 5), (5, 7), (11, 13), (17, 19), (29, 31), (41, 43), ....

so that the primes 3, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43, ... are generated in this way.

It turns out that the prime 23, which is missing from (2), is generated by the "Cousin Primes", which are prime pairs of form (p, p +4), or equivalently by p2 = p1 + 4. These include the pair (19, 23), which is how 23 comes to be in this scenario. 37 and 47 also enter this way.

The Factorial Primes FP are of form:

3) FP = n! +/- 1, n positive integer

and they generate 1 + 1 = 2, 2 + 1 = 3, 3! + 1 = 6 + 1 = 7, 3! - 1 = 6 - 1 = 5, etc.

Osher Doctorow

---- mdoctorow@ca.rr.com wrote:
> 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)