A question on pure math research for integer factorization

I'm interested in integer factorization for a project I'm working on (no it doesn't involve RSA) and I've notice that the majority of papers seemed to be focused on algorithms. Applications aside, I haven't seen a great deal of recent theoretical work on factorization.

If someone could link a few papers focusing on the theory of factorization instead of algorithms, then I'd be most appreciative.

Edit:

First, it seems I accidently deleted the well thought out and rigorous reply. I am truly sorry and I hope that it can be recovered. My apologizes.

Second, let me refine my query a bit. I am research patterns of integer factorization looking for a general recursive relationship: give an integer factorization M for an arbitrary integer N, what can we say about N+1's factorization?

Most IF research is focused on finding particular IFs to break the RSA cryptosystem. I could care less about the speed of factorization as I'm looking for general results to help me prove something related to IF.

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I'm sorry I think I deleted your comment. I feel terrible about it – Charles Hoskinson Mar 23 '13 at 23:39

1 Answer

Given the factorization of $n$, we can say next-to-nothing about the factorization of $n+1$. We can say no prime that divides $n$ also divides $n+1$; we can say that if $2$ doesn't divide $n$, then it does divide $n+1$; that's about it. If, say, $n$ is prime, $n+1$ could have hundreds of factors, or it could be twice a prime.

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That's the point. There has to be some body of research attempting to say something – Charles Hoskinson Mar 24 '13 at 18:38
@CharlesHoskinson, no, there really doesn't. The idea you have described does not go anywhere. Meanwhile, you mentioned, in edit, something you want to prove about integer factorization. Why do you believe that item to be true? – Will Jagy Mar 24 '13 at 19:02
first, please lay off the condescending attitude. There has been over 2000 years of research studying IF. Yes you can say some things. Second, I have significant numerical evidence to suggest what I'm trying to prove is true (up to 4 X 10^18). Third, most research papers study efficient algorithms thus making it difficult to find papers focusing on certain patterns outside the work of Hardy and Tao. – Charles Hoskinson Mar 25 '13 at 7:26
Furthermore, prime numbers do not exhibit a random distribution therefore it stands to reason that factorization patterns are also not random. I understand the difficulty in saying interesting things about them, yet one really could if he wanted. If you don't know, then just say so. Please don't just shoot me down and offer no help. – Charles Hoskinson Mar 25 '13 at 7:30
Charles, this would go better if you would lay your cards on the table and tell us what you know instead of hinting at it obscurely. It's hard to offer you help when you clearly have something specific in mind but refuse to let us in on it. – Gerry Myerson Mar 25 '13 at 12:16