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We give an extension of a theorem of Cesaro from the rational integers to the ring of integers of an arbitrary number field. This extension is used to generalize Pillai's function to number fields.

A method is given for generating families of non-congruent numbers with arbitrarily many prime factors. We then use this method to construct an infinite set of new families of these numbers with prime factors of the form 8 k + 3 .

In this paper we investigate the distribution moidulo 1 of the sequence of vectors ( p α 1 ,…,p α 1 ), where α 1 ,…,α 1 are fixed real numbers lying in (0, 1) and p runs over the set of prime numbers.