Introduction. Some time ago the author was asked by Professor D. N. Lehmer if there was anything known about the representation of an integer A in the form h -¿ In, i-X where all the prime factors of each A,-are of a given form. A search of the literature seemed to indicate that various theorems had been conjectured but none actually proved.f For example, L. Euler stated without proof that every integer of the form 47+2 is a sum of two primes each of the form 47 + 1. Even the weaker statement

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... at every integer of the form 4/+2 is a sum of two integers which have all their prime factors of the form 4/+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 BrunJ to prove a general theorem and from this we deduce Theorems 4 and 5 below.