# Search Results

A graph*H* divides a graph*G*, written*H*|*G*, if*G* is*H*-decomposable. A graph*G* without isolated vertices is a greatest common divisor of two graphs*G*
_{1} and*G*
_{2} if*G* is a graph of maximum size for which*G*|*G*
_{1} and*G*|*G*
_{2}, while a graph*H* without isolated vertices is a least common multiple of*G*
_{1} and*G*
_{2} if*H* is a graph of minimum size for which*G*
_{1}|*H* and*G*
_{2}|*H*. It is shown that every two nonempty graphs have a greatest common divisor and least common multiple. It is also shown that the ratio of the product of the sizes of a greatest common divisor and least common multiple of*G*
_{1} and*G*
_{2} to the product of their sizes can be arbitrarily large or arbitrarily small. Sizes of least common multiples of various pairs*G*
_{1},*G*
_{2} of graphs are determined, including when one of*G*
_{1} and*G*
_{2} is a cycle of even length and the other is a star.

## Abstract

We prove that for any positive real number

## Abstract

For any positive integer n let F_{n} be the n-th Fibonacci number. Given positive integers a and b, we study the size of the greatest common divisor of F_{n } + a and F_{m } + b for varying positive integers m and n.

## Abstract

We study those functions that can be written as a finite sum of periodic integer valued functions. On ℤ we give three different
characterizations of these functions. For this we prove that the existence of a real valued periodic decomposition of a ℤ
→ ℤ function implies the existence of an integer valued periodic decomposition with the same periods. This result depends
on the representation of the greatest common divisor of certain polynomials with integer coefficients as a linear combination
of the given polynomials where the coefficients also belong to ℤ[*x*]. We give an example of an ℤ → {0, 1} function that has a bounded real valued periodic decomposition but does not have a
bounded integer valued periodic decomposition with the same periods. It follows that the class of bounded ℤ → ℤ functions
has the decomposition property as opposed to the class of bounded ℝ → ℤ functions. If the periods are pairwise commensurable
or not prescribed, then we get more general results.