On a Theorem on sums of the form 1+2^(2^n)+2^(2^n+1)+...+2^(2^n+m) and a result linking Fermat with Mersenne numbers

TL;DR

This paper generalizes a divisibility property of certain exponential sums, linking Fermat and Mersenne numbers, and provides explicit constructions for numbers divisible by given odd integers.

Contribution

It proves that for any odd integer greater than 1, there are infinitely many sums of a specific form divisible by it, and establishes a connection between Fermat and Mersenne numbers.

Findings

Existence of infinitely many sums divisible by any odd N>1

Explicit construction of n and m for divisibility

Linking properties of Fermat and Mersenne numbers

Abstract

In his book "250 Problems in Elementary Number Theory", W.Sierpinski shows that the numbers 1+2^(2^n)+2^(2^n+1) are divisible by 21; for n=1,2,.... In this paper, we prove a similar but more general result.Consider the natural numbers of the form I(n.m)= 1+2^(2^n)+2^(2^n+1)+...+2^(2^n+m).In Theorem 1 we prove that for every odd integer N greater than 1, there exist infinitely many natural numbers n and m such that the integers I(n.m) are divisible by N. We give an explicit construction of the numbers n and m, for a given N. As an example, when N=31, and with n=4k and m=94+124i, the numbers I(n,m) are divisible by 31. A similar example is offered for N=(31)(7)=217. In Theorem 2, we prove a result pertaining to Mersenne numbers.There are also three Corollaries in this work, one of which deals with Fermat numbers.