Let denote the number of integers for which , let denote the number of for which and let . Carmichael conjectured that for all (that is, ); Sierpiński conjectured that for every integer , there does exist with .
In this delightful paper, Ford proves Sierpiński’s long elusive conjecture, even proving the surprising result that for , whenever . He had already proved extraordinarily strong estimates for in [Ramanujan J. 2, 67-151 (1998; Zbl 0914.11053)]. The author and S. Konyagin [Number Theory in Progress, Vol. 2, 795-803 (1999; Zbl 0931.11032)] had already proved Sierpiński’s conjecture for even , but the odd case seems to be considerably more difficult.
The starting point for Ford’s method is an easy proof of Sierpiński’s conjecture under the assumption of the prime triplets conjecture; for if and is a large prime such that is prime for dividing only when or , then . Of course we do not know whether the prime triplets conjecture is true, but we do know, from sieve methods, that we can determine -tuples of “almost primes”, that is integers with few and only large prime factors. Thus Ford suitably modifies the above construction to incorporate such -tuples. This requires considerable ingenuity and technical progress, making this proof an impressive achievement.