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 X-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 X-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 w4ose exponent is required, and both a and b are Ak-numbers, that is, having no factor larger than Pk, 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 Ak-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. 1. The proof of primality, or the factorisation, of large integers has been a subject of major interest to mathematicians and others for centuries. It always remains a difficult problem because any method that becomes available is always pushed speedily to its limits. The straightforward method for deciding both versions of the problem completely is to use the fact that the number N is either a prime, or has a factor not exceeding VN. We may therefore try to divide N by each prime up to VN; this eventually solves the problem in a number of operations of maximum order VNslightly less if a list of primes is available. If we have no list of primes we can try by using all odd numbers, possibly excluding those which themselves have an obvious small factor. However the number of operations is still basically of order IN. There are other methods, for example involving quadratic forms N = Ax2 + By2, x, y integers, that depend on trials with a similar number of operations. Such methods, with number of operations of basic order IN, although this may, in a particular case of actual factorisation, turn out to be a considerable overestimate, Received August 15, 1974. AMS (MOS) subject classifications (1970). Primary 1OA10, 10A25; Secondary 10-04.
[1]
D. H. Lehmer.
Some new factorizations of $2^n \pm 1$
,
1933
.
[2]
D. H. Lehmer.
On the Converse of Fermat's Theorem
,
1936
.
[3]
D. H. Lehmer.
A factorization theorem applied to a test for primality
,
1939
.
[4]
de Ng Dick Bruijn.
On the number of positive integers $\leq x$ and free of prime factors $>y$
,
1951
.
[5]
J. L. Selfridge,et al.
Some factorizations of 2ⁿ±1 and related results
,
1967
.
[6]
J. Brillhart,et al.
Corrigendum: “Some factorizations of 2ⁿ±1 and related results”
,
1967
.
[7]
A. E. Western,et al.
Tables of indices and primitive roots
,
1968
.
[8]
D. G. Hazlewood.
On Integers all of whose Prime Factors are Small
,
1973
.