Q) Prove or disprove: there exist infinitely many composite numbers of the form , where is any positive integer.
I am not sure how to approach this. I beleive that the statement is true, but I am not sure how to approach this proof. My first thought is to do a proof by contradiction by assuming I can list all such composite number. However, I cannot find a way to arrive at a contradiction.
Some direction would be appreciated.
So, the identity with tells us a number of the form is composite whenever n is composite. Since there are infinitly many composite numbers, there must be infinitly many composite numbers of the form .
Is this sufficient? Do I need to specify the exponent n mentioned above has to be the product of two integers greater than one?
Thanks for the help
I think it's sufficient, but to make it more formal, you can consider the set of positive composite integers, call it A, then consider the bijection from A -> B mapping x in A to 2^x-1 in B, and conclude that B is countably infinite. (B is a subset of the set of composite Mersenne numbers.)