Jon Perry wrote:

> I know of the result that says if 30% of N-1 is factored,

> then it becomes more possible to factor N (excuse my lack

> of knowledge in this area - my research has not yet touched

> upon it fully).

I think you may be confusing factorization of N with primality

testing of N. I deduce this from the mention of "30%".

If a number N is believed to be prime, and one has at least 30%

of the factorization of N-1, it is then straightforward to prove

whether N is prime.

With this sole exception (proving that the prime factorization

of N is just N) I know of nothing that makes factorization of

N any easier if one knows the partial factorization of N-1.

If it did, we would find Fermat numbers much easier to factor

than they actually are: we know the complete factorization of

N-1 in that case.

> Is there a similar result that allows N to be factored using

> knowledge of other surrounding n's, e.g. N-k, .., N-1, N+1,

> ...N+j?

Yes, subject to the proviso that N is prime and it is wished to

prove this. The factorization of N+1 can be used in a similar

manner, as can the factorization of quantities such as the

algebraic factors of N^6-1 (of which N-1 and N+1 are just two).

There are no known ways of exploiting the factorizations of

N \pm j to find the factors of N.

Paul