Let $n$ be any given natural number. Let $p$ be the very next prime greater than $n$. Let $b$ be the bound for the prime gap above $n$. Here, the bound is strictly the limit from $n$ to $p$, meaning that $n$+$b$ is greater than or equal to $p$. $n$+$b$ will never be less than $p$. Most often, $n$+$b$ will be greater than p. Sometimes, $n$+$b$ will be equal to p.
I was told that Legendre's conjecture implies that the prime gap above any natural number $n$ is bounded by the product of a constant and the $\sqrt n$. Now, I need to reconsider this. I have discovered a new function for what Legendre's conjecture implies.
Actually, I have strong reason to think that this factor by which the constant is multiplied is not $\sqrt n$.
The question is this: Is it possible for this factor to be a whole number that is less than $\sqrt n$? Is this a possibility which would be consistent with all the leading theories?
Is the following possible?:
The factor could be less than the square root of $n$. The bound is 2 multiplied by a factor. 2 is the constant. The factor is not the square root of $n$. It is a whole number that is less than the square root of $n$.
I believe this.
I also have strong reasons for thinking the prime gap should not be so large, namely Cramér's conjecture, which is a heuristic argument that the prime gap should be $O((\log n)^2)$.
But so far the best we can do is show the gap is $O(\sqrt{n} \log n)$, assuming the Riemann Hypothesis (this is also due to Cramér).