ON THE DISTRIBUTION OF M-TUPLES OF B-NUMBERS

Abstract

In the classical sense, the set B consists of all integers which can be written as a sum of two perfect squares. In other words, these are the values attained by norms of integral ideals over the Gaussian field Q(i). G. J. Rieger (1965) and T. Cochrane, R. E. Dressler (1987) established bounds for the number of pairs (n; n + h), resp., triples (n; n + 1; n + 2) of B-numbers up to a large real parameter x. The present article generalizes these investigations into two directions: The result obtained deals with arbitrary M-tuples of arithmetic progressions of positive integers excluding the trivial case that one of them is a constant multiple of one of the others. Furthermore, the estimate applies to the case of an arbitrary normal extension K of the rational field instead of Q(i).