What is the largest known prime gap, and why is there so much conflicting information??

Question

I find consistently conflicting information online.

On the one hand there's all these reports of the largest prime gap between all numbers proven to be < 246.

On the other hand, I read reports of the largest known prime number gap being 1476.

On the other hand, I find all kind of different reports, such as this one: http://sweet.ua.pt/tos/gaps.html where there are many prime numbers with gaps > 246.

What am I failing to understand?

Thanks!

EDIT: There's many good answers on this page. Thanks so much you guys! I only chose dalastboss's because I found it easiest to grasp. It was 'dumbed down' enough for me :)

Answer 1

For any $ n > 1 $, there is a prime gap of size at least $ n $. The results you read probably meant that it's been proven that the least gap which occurs infinitely often is less than 246. That is $ \liminf\limits_{n \rightarrow \infty} \; (p_{n+1} - p_n) < 246 $. $\liminf\limits_{n \rightarrow \infty} \; (p_{n+1} - p_n) =_? 2$ is the twin prime conjecture.

Answer 2

For every number $n\in\mathbb{N}$ that you can think of, I can give you a sequence of $n-1$ consecutive numbers, none of which is prime.

There you go: $n!+2,n!+3,\dots,n!+n$.

So there is no finite bound on the gap between two consecutive primes.

Answer 3

User barak manos provides a nice proof for the divergence of the limit superior of the sequence of prime gaps $g_n$, where, if $p_k$ is the $k^\text{th}$ prime,

$$g_n = p_{n+1}-p_n$$

Even stronger results have been proven. For example, for any (arbitrarily large if you like) $c \in \mathbb{R}$, there exist infinitely many $n \in \mathbb{N}$ such that the prime gap $g_n$ satisfies

$$g_n > \frac{c\log n \log\log n \log \log\log \log n }{\left(\log\log\log n\right)^2}$$