Introducing Fermat Sequences

Abstract

This paper focuses on simple observation of Fermat s factorization method (FFM) applied on the first 5 × 106 odd number. The number of iteration used in the FFM is related to the factor that the method found for each odd number. A new collection of sequences or sets of numbers (we called it Fermat-d sequences) can be generated based on the factor that we found using this method. For example, the first member (Fermat-1) in this collection of sequences is odd prime number sequence, which has factor–of course–of 1 and has highest number of iteration in the FFM, which is ~half of the number itself. The next member of these sequences (Fermat-3, Fermat-5, and so on) are always start from d2, continue with arithmetic progression of 2d, but, with additional random jumps related to the prime number distribution. We present the first exploration of these new sequences (not registered in oeis.org yet) and the beauty of the location of these sequences in the Ulams and Sacks spiral.