A Survey On Practical Numbers

. A positive integer m is said to be practical if every integer n 2 (1; m) is a sum of distinct positive divisors of m: In this paper we give an equivalent definition of practical number, and describe some arithmetical properties of practical numbers showing a remarkable analogy with primes. We give an improvement of the estimate of the gap between consecutive practical numbers and prove the existence of infinitely many practical numbers in suitable binary recurrence sequences, including the sequences of Fibonacci, Lucas and Pell. 1. Introduction. A positive integer m is said to be practical (see [11]) if every n with 1 n m is a sum of distinct positive divisors of m: Several authors dealt with some aspects of the theory of practical numbers. P. Erdos [3] in 1950 announced that practical numbers have zero asymptotic density. B. M. Stewart [12] proved the following structure theorem: an integer m 2; m = p ff 1 1 p ff 2 2 \Delta \Delta \Delta p ff k k ; with primes p 1 ! p 2 !..