
AdditiveSubtractive Generation of Primes
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 "nearsquare primes" (see for example Wikipedia online or Wolfram/Eric Weisstein online articles on "NearSquare 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

