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In his book "250 Problems in Elementary Number Theory", W.Sierpinski showsthat the numbers 1+2^(2^n)+2^(2^n+1) are divisible by 21; for n=1,2,.... Inthis paper, we prove a similar but more general result.Consider the naturalnumbers of the form I(n.m)= 1+2^(2^n)+2^(2^n+1)+...+2^(2^n+m).In Theorem 1 weprove that for every odd integer N greater than 1, there exist infinitely manynatural numbers n and m such that the integers I(n.m) are divisible by N. Wegive an explicit construction of the numbers n and m, for a given N. As anexample, when N=31, and with n=4k and m=94+124i, the numbers I(n,m) aredivisible 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 threeCorollaries in this work, one of which deals with Fermat numbers.

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