Products of two proportional primes

Abstract

In RSA cryptography numbers of the form [Formula: see text], with [Formula: see text] and [Formula: see text] two distinct proportional primes play an important role. For a fixed real number [Formula: see text] we formalize this by saying that an integer [Formula: see text] is an RSA-integer if [Formula: see text] and [Formula: see text] are primes satisfying [Formula: see text]. Recently Dummit, Granville and Kisilevsky showed that substantially more than a quarter of the odd integers of the form [Formula: see text] up to [Formula: see text], with [Formula: see text] both prime, satisfy [Formula: see text]. In this paper, we investigate this phenomenon for RSA-integers. We establish an analogue of a strong form of the prime number theorem with the logarithmic integral replaced by a variant. From this we derive an asymptotic formula for the number of RSA-integers [Formula: see text] which is much more precise than an earlier one derived by Decker and Moree in 2008.