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

  • Thread starter maris205
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In summary, there are large gaps in the number of prime numbers between p and p^2, but it's possible to conjecture that the gaps grow slowly and there is a small example to find by inspecting a table of primes.
  • #1
maris205
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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.
 
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  • #2
maris205 said:
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.
 
  • #3
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.
 
  • #4
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.
 
  • #5
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.
 
  • #6
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 occurance 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).
 

1. What is the significance of studying the gaps between p and p^2?

The gaps between p and p² provide important insights into the relationship between prime numbers and their squares. It can help us better understand the distribution and patterns of prime numbers, which has been a subject of interest for mathematicians for centuries.

2. How are the gaps between p and p^2 calculated?

The gap between p and p² is simply the difference between the two numbers. For example, if p = 5, then the gap between p and p² is 25 - 5 = 20. This calculation is done for every prime number and its corresponding square to create a sequence of gap values.

3. Do the gaps between p and p^2 follow a specific pattern?

No, the gaps between p and p² do not follow a specific pattern. They appear to be randomly distributed, with some gaps being very small and others being very large. This is one of the reasons why studying these gaps is important, as it can help us uncover new patterns and relationships.

4. Are there any practical applications for understanding the gaps between p and p^2?

While the study of these gaps may not have direct practical applications, it has implications for various fields such as coding theory, cryptography, and number theory. Additionally, understanding the distribution of prime numbers and their squares can help us improve algorithms and encryption methods.

5. How does the size of the gap between p and p^2 change as p increases?

The size of the gap between p and p² tends to increase as p increases. This is because as p gets larger, its square also gets larger at a faster rate. However, there are exceptions to this trend and some small primes can have large gaps between them and their squares. This irregularity is part of what makes the study of prime numbers and their gaps so fascinating.

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