Is the set of prime number finite? if?

[Shad0w7]

Let's say I have this statement. {a^p | p is prime and p < N}

a is considered a string so

so a^2 = aa, a^3 = aaa and so on...

anyway, in this case, since it says that p< N, then is mean that p will be finite right??

 

[HallsofIvy]

First, let me point out that the answer to the question asked in the title, "is the set of prime numbers finite", is "NO"- the set of all prime numbers is infinite- that proof was given by Euclid, thousands of years ago.

But the answer to the question asked in your text, "Is the set of all numbers of the form $a^p$ where a is a given number and p is a prime number less than N finite" is "YES". In fact, the "prime" part is irrelevant. If a is a fixed number, then the set of all $a^n$, where n is any positive integer less than N, is finite.