There Are Innnitely Many Carmichael Numbers Larger Values Were Subsequently Found

Fermat wrote in a letter to Frenicle, that whenever p is prime, p divides a p?1 ? 1 for all integers a not divisible by p, a result now known as Fermat's `little theorem'. An equivalent formulation is the assertion that p divides a p ? a for all integers a, whenever p is prime. The question naturally arose as to whether the primes are the only integers exceeding 1 that satisfy this criterion, but Carmichael Ca1] pointed out in 1910 that 561 (= 3 11 17) divides a 561 ? a for all integers a. In 1899, Korselt Ko] had noted that one could easily test for such integers by using (what we will call) Korselt's criterion: n divides a n ? a for all integers a if and only if n is squarefree and p ? 1 divides n ? 1 for all primes p dividing n. In a series of papers around 1910, Carmichael began an in-depth study of composite numbers with this property, which have become known as Carmichael numbers. In Ca2], Carmichael exhibited an algorithm to construct such numbers and stated, perhaps somewhat wishfully, that \this list (of Carmichael numbers) might be indeenitely extended". Indeed, until now, no one has been able to prove that there are innnitely many Carmichael numbers, though it has long seemed highly likely. In 1939 Chernick noted that if p = 6m+1; q = 12m+1 and r = 18m+1 are all prime then pqr is a Carmichael number. According to Hardy and Littlewood's widely believed prime k-tuplets conjecture, these should simultaneously be prime innnitely often, which would tell us that there are innnitely many Carmichael numbers. the other hand, numerous authors have supplied upper bounds for C(x), the number of Carmichael numbers up to x, the best being ((PSW], though also see Po]) C(x) x 1?f1+o(1)g log log log x= log log x 1 for x ! 1. We believe that this upper bound probably gives the true size of C(x). Our belief can be justiied by the heuristic argument in Po], which is based on ideas of Erd} os Er2]. In this paper we show that C(x) > x for all large x and some positive constant. In particular, we may take = 2=7. A precise upper bound for allowable values of in our theorem depends on two other constants that appear in analytic number theory. We now describe …