Primality proofs with elliptic curves: on the distribution and the expected value of the
gain during downrun

Abstract. In this article, we give heuristic approximation for the distribution, and the expected value of the gain during one step of the downrun part of the elliptic curve primality proving method.

Key words and phrases. Elliptic curve primality proving, elliptic curve order, primes in short intervals, smooth numbers in short intervals, Dickman function.