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The following question is part of a question I had to solve for my masters exam.

Consider $S=\{s_n =p_{n+1}- p_n \mid n \in \mathbb{N}\}$. Prove that sup $S=\infty$.

Intutively I guessed it and Tried using prime number theorem and L Hopital rule but it comes out to be very lenghty and still indeterminant.

So, I am looking for altenative rigorious method.

Kindly tell how to prove it.

Thanks!!

Clyde Kertzer
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    You just need to show that you can find consecutive primes with arbitrarily large gaps. A standard way to show this would be to consider the numbers near factorials $n!$ – player3236 Nov 04 '20 at 13:59
  • The answers here might help: https://math.stackexchange.com/questions/2311652/im-trying-to-find-the-longest-consecutive-set-of-composite-numbers?noredirect=1&lq=1 – player3236 Nov 04 '20 at 14:01
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    For every possible distance between successive primes $k=p_{i+1}-p_i$, the factorial $(k+1)!$ has at least $k$ composite numbers in a row in the interval $[(k+1)!+2,(k+1)!+k+1]$. This same approach can be applied using a number like $(k+1)#$ as well, with a small amount of logic to obtain the result. – abiessu Nov 04 '20 at 14:03

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