Any conclusions about the gaps between p and p^2?

[maris205]

In sieve method, we could get the prime numbers between p and p^2 applying the primes less than prime number p.

Is there any conclusion about the gap of these primes? Some conjectures show the upper bound of primes gaps before p is g(p)< ln(p)^2 or g(p)<p^(1/2) (If RH is true).

But here we just consider the primes between p and p^2, which are filtered by determinate primes. So I want to know whether we could get some easy conclusions about their gaps by elementary number theory. For example, could we prove the gap of the primes between p and p^2 is less than p? My calculation shows it’s true.

 

[shmoe]

Is there any conclusion about the gap of these primes? Some conjectures show the upper bound of primes gaps before p is g(p)< ln(p)^2 or g(p)<p^(1/2) (If RH is true).

RH would imply g(p)=O((log p)*p^(1/2)), not p^(1/2).

I don't know whose conjecture your g(p)< ln(p)^2 is referring to? There's Cramer's that says the lim sup of g(p)/log(p)^2 is 1 but this doesn't imply your inequality.

But here we just consider the primes between p and p^2, which are filtered by determinate primes. So I want to know whether we could get some easy conclusions about their gaps by elementary number theory. For example, could we prove the gap of the primes between p and p^2 is less than p? My calculation shows it’s true.

What's a "determinate prime"? Knowing the largest gap between p and p^2 is p implies the largest gap less than p is p^(1/2) (just consider a prime larger than sqrt(p) and apply this again, and repeat). Since this is quite a large leap from current results (and stronger than what RH implies) you're going to have to provide something stronger than "My calculation show's it's true" before I come near believing you can prove this.

 

[matt grime]

p=13, p^2=169, 113 and 127 are prime, no number between them is prime (5|115, 9|117, 7|119, 11|121, 3|123, 5|125), and 127-113=14>13.

Thankfully there is a small example to find by inspecting a table of primes. I doubt that this is is an isolated example.

 

[shmoe]

It would seem suprising if that wasn't isolated (barring including consecutive primes like 7,11 where only one lies between 3 and 3^2). See

http://primes.utm.edu/notes/GapsTable.html

The numerical evidence points to gaps quite a bit smaller than we can prove at this point.

 

[matt grime]

That link brings up one question: which definition of difference is the OP using: p-q or p-q-1? The latter makes my counter example false, but I assumed the former, naively.

The gaps are indeed far smaller than I anticipated.

 

[shmoe]

p-q or p-q-1, either way a finite number of counter examples (or none) seems likely.

If p(g) is the prime following the first occurrence of a gap of length at least g, it's conjectured that log p(g)~sqrt(g). The maximum gap length seems to grow very slowly. Given the average gap between primes less than x is log(x), it's maybe not too suprising.

The basic proof that there are arbitrarily large gaps doesn't usually stress just how big the number constructed is. A gap of length n-1 following n!+1 is pretty short considering just how large n! is (of course the actual gap here may be bigger, you'd take the largest prime less than n!+2, etc).