# Another algebra problem about prime and induction

1. Sep 21, 2010

### kntsy

1. The problem statement, all variables and given/known data

prove by induction that the $n^{\text th}$ prime is less than $2^{2^{\text n}}$

2. Relevant equations
hint:assume it is correct for all $n \leq k$, and then compare $p_{k+1}$ with $p_{1}p_{2}..........p_{k}+1$

3. The attempt at a solution
is $p_{k+1}$ smaller/greater than $p_{1}p_{2}..........p_{k}+1$ so that i can extend the use of inequality?
I attempt to use euclidean algorithm but do not know where to use.
Thank you.

2. Sep 21, 2010

### Office_Shredder

Staff Emeritus
If $$p_{k+1}$$ is larger than the suggested number, you should be able to prove that it has no prime divisors which is a contradiction