On factorisation, with a suggested new approach

Abstract:

This paper gives a brief survey of methods based mainly on Fermat’s Theorem, for testing and establishing primality of large integers. It gives an extension of the Fermat-Lucas-Lehmer Theorems which allows us to establish primality, or to factorise composites, in cases where the Carmichael $\lambda$-exponent is known (or a multiple or submultiple of it, by a moderate factor). The main part of the paper is concerned with describing a method for determining the $\lambda$-exponent in cases where the Fermat test is not satisfied. This method is a variation of A. E. Western’s method for finding indices and primitive roots, based on congruences $N = a + b$, where N is the number whose exponent is required, and both a and b are ${A_k}$-numbers, that is, having no factor larger than ${p_k}$, the kth prime. The most onerous problem lies in the finding of a sufficient number of congruences (at least k) and in the choice of a suitable value of k. The determination of the approximate number of ${A_k}$-splittings available is considered, to allow an estimate of the amount of labour (human or electronic) needed to be made. The final suggestion, rather inconclusive, is that the method has possibilities worth exploring further and may be as economical, after development, as existing methods, and possibly more so when N is large.