where all the prime factors of each hi are of a given form. A search of the literature seemed to indicate that various theorems had been conjectured but none actually proved.t For example, L. Euler stated without proof that every integer of the form 4j+2 is a sum of two primes each of the form 4j+ 1. Even the weaker statement that every integer of the form 4j+2 is a sum of two integers which have all their prime factors of the form 4j+ 1 has not yet been proved. In view of the absence of any definite results in the literature it seems worthwhile to point out that some very interesting theorems can be obtained in an elementary way. This is done in Part I of this paper and the results are summarized in Theorems 1, 2, and 3 below. In Part II we use the method of Viggo Brunt to prove a general theorem and from this we deduce Theorems 4 and 5 below.