Divisor Problems in Special Sets of Positive Integers

Extremal Problems for Subset Divisors

Let A be a set of n positive integers. We say that a subset B of A is a divisor of A, if the sum of the elements in B divides the sum of the elements in A. We are interested in the following extremal problem. For each n, what is the maximum number of divisors a set of n positive integers can have? We determine this function exactly for all values of n. Moreover, for each n we characterize all s...

On Integers Expressible by Some Special Linear Form

Let E(4) be the set of positive integers expressible by the form 4M −d, where M is a multiple of the product ab and d is a divisor of the sum a + b of two positive integers a, b. We show that the set E(4) does not contain perfect squares and three exceptional positive integers 288, 336, 4545 and verify that E(4) contains all other positive integers up to 2 · 109. We conjecture that there are no...

Consecutive Integers with Equally Many Principal Divisors

Classifying the positive integers as primes, composites, and the unit, is so familiar that it seems inevitable. However, other classifications can bring interesting relationships to our attention. In that spirit, let us classify positive integers by the number of principal divisors they possess, where we define a principal divisor of a positive integer n to be any prime-power divisor pa|n which...

ASYMPTOTIC FORMULAE CONCERNING ARITHMETICAL FUNCTIONS DEFINED BY CROSS-CONVOLUTIONS, I. DIVISOR-SUM FUNCTIONS AND EULER-TYPE FUNCTIONS By László Tóth (Cluj-Napoca)

Abstract. We introduce the notion of cross-convolution of arithmetical functions as a special case of Narkiewicz’s regular convolution. We give asymptotic formulae for the summatory functions of certain generalized divisor-sum functions and Euler-type functions related to cross-convolutions and to arbitrary sets of positive integers. These formulae generalize and unify many known results concer...

Quarternions and the Four Square Theorem

The Four Square Theorem was proved by Lagrange in 1770: every positive integer is the sum of at most four squares of positive integers, i.e. n = A2 + B2 + C2 + D2, A, B, C, D ∈ Z An interesting proof is presented here based on Hurwitz integers, a subset of quarternions which act like integers in four dimensions and have the prime divisor property. It is analogous to Fermat’s Two Square Theorem,...

Sets of Integers and Quasi–integers with Pairwise Common Divisor