The Sum of Binomial Coefficients and Integer Factorization

The combinatorial sum of binomial coe cients  n i r (a) := X k⌘i (mod r) ✓ n k ◆ a k has been studied widely in combinatorial number theory, especially when a = 1 and a = 1. In this paper, we connect it with integer factorization for the first time. More precisely, given a composite n, we prove that for any a coprime to n there exists a modulus r such that the combinatorial sum has a nontrivial greatest common divisor with n. Denote by FAC(n, a) the least r. We present some elementary upper bounds for it and believe that some bounds can be improved further since FAC(n, a) is usually much smaller in the experiments. We also proposed an algorithm based on the combinatorial sum to factor integers. Unfortunately, it does not work as well as the existing modern factorization methods. However, our method yields some interesting phenomena and some new ideas to factor integers, which makes it worthwhile to study further.