Products of three pairwise coprime integers in short intervals

Abstract

The existence of products of three pairwise coprime integers is investigated in short intervals of the form (x, x + x 1 2 ]. A general theorem is proved which shows that such integer products exist provided there is a bound on the product of any two of them. A particular case of relevance to elliptic curve cryptography, where all three integers are of order x 1 3 , is presented as a corollary to this result.