Tarski asked whether the arithmetic identities taught in high school are complete for showing all arithmetic equations valid for the natural numbers. The answer to this question for the language of arithmetic expressions using a constant for the number one and the operations of product and exponentiation is affirmative, and the complete equational theory also characterises isomorphism in the ty...

We show that the number of maximal sum-free subsets of {1, 2, . . . , n} is at most 2. We also show that 2 is an upper bound on the number of maximal product-free subsets of any group of order n.

It is shown that the set of decimal palindromes is an additive basis for the natural numbers. Specifically, we prove that every natural number can be expressed as the sum of forty-nine (possibly zero) decimal palindromes. 1. Statement of Result Let N ..= {0, 1, 2, . . .} denote the set of natural numbers (including zero). Every number n 2 N has a unique decimal representation of the form

A simple undirected graph H is called a sum graph if there is a labeling L of the vertices of H into distinct positive integers such that any two vertices u and v of H are adjacent if and only if there is a vertex w with label L(w) = L(u) + L(v). The sum number (G) of a graph G = (V; E) is the least integer r such that the graph H consisting of G and r isolated vertices is a sum graph. It is cl...

The sum in the title is a rational multiple of π for all integers n = 2, 3, 4, . . . for which the sum converges absolutely. This is equivalent to a celebrated theorem of Euler. Of the many proofs that have appeared since Euler, a simple one was discovered only recently by Calabi [4]: the sum is written as a definite integral over the unit n-cube, then transformed into the volume of a polytope ...

Generalizing a classical one-variable theorem of Bohr, we show that if an n-variable power series has modulus less than 1 in the unit polydisc, then the sum of the moduli of the terms is less than 1 in the polydisc of radius 1/(3 √ n ). How large can the sum of the moduli of the terms of a convergent power series be? Harald Bohr addressed this question in 1914 with the following remarkable resu...

We show that for almost all choices of parameters, the elliptic subset sum pseudorandom number generator produces a sequence of uniformly distributed pseudorandom numbers. The result is useful for both cryptographic and Quasi Monte Carlo applications and relies on bounds of exponential sums.