Additive representations of primes and the distribution of special primes
by
Takashi Agoh
Science University of Tokyo

Let p_i be the ith prime, thus p₁=2,  p₂=3,   p₃=5, and so on. For a positive integer m, we put

Nm= m Õ i=1  pi

and C_i=N_m/p_i  (i=1, 2, ..., m). It is known that a positive integer n with

n <= m å i=1  Ci

and (n, N_m)=1 can be expressed as ( * )  n=K₁+K₂+ ... +K_m, where each K_i is an integer such that |K_i| < N_m and the set of prime divisors of K_i is exactly {p₁, p₂, ...p_m}\{p_i}. Conversely, if an integer n with p_m <= \surd{ n } < p_m+1 is given in the form ( * ), then we can see at a glance that n is a prime. So the expression ( * ) of n gives a proof of primality.

The main purpose of this talk is to deduce some additive representations of primes which are similar to ( * ), and to discuss the distribution of some special primes by making use of these representations. Further we would like to offer certain open questions relating to these topics.

Date received: April 19, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacf-38.