A survey of recent results in elementary number theory is presented in this paper. Special attention is given to structure and asymptotic properties of certain families of positive integers. In particular, a conjecture on complete sequences of Burr, Erdős, Graham and Wen-Ching Li is amended.

As a generalization of the equation 0(x) + 0(&) = 0(x + /t), 0-partitions and reduced 0-partitions and reduced 0-partitions of positive integers were considered by Patricia Jones [1]. That is., n = ax + —\-at is a 0-partition if i> 1 and <j)(n) = §{ax) + '' ' + 0 ( a

) + f(3a) + · · · + f(ℓ a) |> M? The answer to the first question is yes. In words, every two-coloring of the positive integers has unbounded discrepancy, taken over the family of arithmetic progressions. Restricting attention to the subset {1, 2, 3,..., n}, we have [1, 2, 3, 4] c n 1/4 ≤ P (n

Many of the attempts to obtain representations for commutative and/or Archimedean semigroups involve using the additive positive integers or subsemigroups of the additive positive integers. In this regard note references [1] , [3] , and [4]. The purpose of this paper is to catalogue the results that

Let n,p,k be three positive integers. We prove that the numbers () n k 3F2(1 − k, −p,p−n; 1,1−n; 1) are positive integers which generalize the classical binomial coefficients. We give two generating functions for these integers, and a straightforward application.

Let a I, a 2i..., am be any finite set of distinct natural numbers, and let b 1, b 2,... be the sequence formed by all those numbers which are divisible by any of a 1, a 2,..., am. This sequence has a density in the obvious sense and we denote this density by A(a I, a 2i..., am). In fact, I I A(a I, a2,...,a.)=! + a

Abstract not found

A P-set will be defined as a set of positive integers such that if a and b are two distinct elements of this set, ab + 1 is a square. There are many examples of P-sets such as [2, 12] or [1, 3, 8, 120] and even formulas such as

Let Q be a set of primes having relative density 6 among the primes, with 0~6 < 1, and let $(x. y. Q) be the number of positive integers <x that have no prime factors from Q exceeding y. We prove that if y-t cc, then r&x, y, Q) w xp6(u), where u = (log x)/(log y), and ps is the continuous

Abstract The main purpose of this paper is using the elementary method and analytic method to study the asymptotic properties of the integer part of the k-th root positive integer, and give two interesting asymptotic formulae.