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!

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 $2=p_1<p_2<\dotsb<p_n<\dotsb$. 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!

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 suppose I should close the question in a few minutes. Thanks to all who put effort!

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!