TITLE: Mathematical induction vis-a-vis primes
QUESTION [5 upvotes]: One of the most used proof-techniques is mathematical induction, and one of the oldest subjects is the study of prime numbers. Thanks to Euclid, we can consider the primes as a infinite monotone sequence $2=p_1<p_2<\cdots<p_n<\cdots$. But, knowing the prime $p_n$ does not tell us the exact location of the next.
My inquiry here is this:

Question. What theorems/results do you know where induction is done on a formula/statement involving primes? In the sense that you move from one prime to the next, inductively. Please include reference.

UPDATE. It seems that the question has confused more people that I thought it would. For this reason, I don't mind if the editors decide to close it. Thanks to all who put effort!

REPLY [2 votes]: Two examples which come to my mind are:

$(\forall n \in \mathbb{N})$ $\, \,$ $p_{n+1}<2^{2^{n}}$.
For every $n \in \mathbb{N}_{\geq 12}$, $\,$ $p_{n}>3n$.

By the way, Erdös's proof of Bertrand's postulate is not by induction (it depends on some results which can be proven via mathematical induction, but that's a different thing): what Erdös actually does in his proof is compare lower bounds for the central binomial coefficients $\binom{2n}{n}$ with some upper ones which he obtains by means of Legendre's formula, the Erdös-Kalmár inequality, and the assumption that there are no primes in $(n,2n]$.