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
| dc.creator | Zelator, Konstantine "Hermes" | |
| dc.date | 2008-06-09 | |
| dc.date.accessioned | 2026-07-07T09:43:25Z | |
| dc.date.available | 2026-07-07T09:43:25Z | |
| dc.description | 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. | |
| dc.description | 12 pages | |
| dc.identifier | https://arxiv.org/abs/0806.1514 | |
| dc.identifier | http://arxiv.org/abs/0806.1514 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/162548 | |
| dc.subject | General Mathematics | |
| dc.title | 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 | |
| dc.type | text |