Infinite number of $k$ satisfying $p_{k+1} - p_{k} > 2$?

Question

I was thinking of how to prove this -

Let $p_{k}$ be the $k$th prime.Then to show that there exists infinite number of $k$ such that $p_{k+1}-p_{k} > 2$.

I was thinking of Twin Prime Conjecture stating that there are an infinite number of primes differing by 2, but how can it help with infinite number consecutive primes differing greater than 2?

I was thinking of going through the method of contradiction but could not proceed

Answer 1

If there are an infinite number of primes congruent to 1 mod 6 (in fact, there are, but we don't even need this), you are done, since there is only one prime congruent to 3 mod 6.

If there are not infinitely many primes congruent to 1 mod 6, then all but finitely many primes are congruent to 5 mod 6, and again you are done.

Answer 2

There are infinitely many odd primes and there is no prime other than $5$ that ends with $5$.

Answer 3

Suppose it's false. Then there are finitely many primes such that $p_{k+1}-p_k>2$, and since there are infinitely many primes, that means that all primes larger than some number $N$ satisfy $p_{k+1}=p_k+2$ (can't be $p_{k+1}=p_k+1$ because even numbers are not primes, except 2).

Thus all odd numbers (except those below $N$) are primes. But among three successive odd numbers larger than $3$, one is necessarily a multiple of $3$, hence not a prime. Hence the hypothesis is false and your sentence is true.