Proof of a stronger statement of the Cramér Conjecture,

[tex]G(X) < log^{2}X[/tex],

for large X where G(X) is the largest prime gap between consecutive primes less than or equal to X:

https://en.m.wikipedia.org/wiki/Cramér%27s_conjecture

Keywords: Prime Number Theorem (PNT)

What constrains the size of G(X)?

The answer is the number, s, of smaller prime gaps between consecutive primes less than or equal to X.

But the average prime gap is size, [tex]\approx log X[/tex], since we exclude all prime gaps of size, G(X).

So, [tex]s * logX \approx X[/tex].

And b is the number of prime gaps of size, G(X).

Therefore, s * log X + b * G(X) = X. But, b << s,

generally.

So, we have,

s * log X + b * G(X) = X which implies

G(X) = (X - s * log X)/b = (X /(b * log X) - s/b) * log X

[tex]\approx[/tex] ([tex]\pi(X)/ b - s /b[/tex]) * log X.

(1). We observe that, [tex]\pi(X) - s \ge 1[/tex],

[tex]\pi(X) - s << \pi(X)[/tex],

and 0 < 1 / b < 1.

(2). Furthermore, we observe that, log X << [tex]\pi(X)[/tex].

We combine the ideas of (1) and (2) to conclude:

[tex]G(X) \approx ( \pi(X) / b - s /b ) * log X < (log X) * log X = log^{2}X[/tex]

Therefore, [tex]G(X) < log^{2}X[/tex] for large X.

Dave.