This is just a thought: if gaps between prime numbers can be arbitrarily large then it should be possible to find infinitely many gaps, such that the product $m=\prod_{n=1}^{N}Pn<P_{N+1}^{2}$, where Pn is the n-th prime. Then m-1, m+1 would have to be twin primes. My question is, what is wrong with this argument:).
Twin prime conjecture and gaps between primes
prime numbersprime-gapstwin primes
Best Answer
The problem with your claim is that it is not possible to find infinitely many inequalities of the sort you claim -- and in particular it does not follow from the previous sentence. Although gaps get arbitrarily large, their size grows much slower than what you need for your argument. The gaps that occur near the $N$th prime will be vastly smaller than needed to make $\prod_{n=1}^N P_n$ less than $P_{N+1}^2$.