Algorithm Available for Factoring Big Fermat Numbers

Abstract

The paper proves that an odd composite integer N can be factorized in at most O( 0.125u (log2N)) searching steps if N has a divisor of the form 2u +1 or 2u-1 with  >1 being a positive integer and u>1 being an odd integer. Theorems and corollaries are proved with detail mathematical reasoning. Algorithms to factorize the kind of odd composite integers are designed and tested with certain Fermat numbers. The results in the paper might be helpful to factorize certain big Fermat numbers.