A finite analogue of the Goldbach problem

Abstract

where 0<cs<,uj (i= 1, c, r), and the ci are uniquely determined (Lemma 3). We say an element i7 is a unit in Rm if (X7, m) = 1. Two elements a, A of Rm are said to be associated if a = 73 where 7 is a unit. On the basis of the above representation, the primes of Rm are simply the elements associated with the (ordinary) primes dividing m. Thus an element (1.2) is a prime of Rm if and only if it is of the form n = qjS [4, p. 294]. In case m is even, we may classify the elements n of Rm into even or odd according as 2 does or does not appear in the factorization (1.2) of n. The basic problems arising in the additive arithmetic of primes in Rm are the following: (1) For what integers m does there exist a number G(m) such that every element of Rm is expressible as a sum of G(m) primes of Rm? For those m for which G(m) exists, what is the minimum value g of G(m)? (2) For a given m, determine H(m), if it exists, such that every element of Rm is expressible as a sum of at most H(m) primes in Rm. The answers to these questions are given in the following two theorems:

Locations

• Proceedings of the American Mathematical Society - View - PDF