Prime Number Gaps - What's the Largest Integer Difference?

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In summary: Thanks for clarifying!In summary, Bertrand's postulate puts a limit on the gap between consecutive primes.
  • #1
rad0786
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Hello everyone,

I'd first like to say that I am uninformed on this subject and that I have a question to the mathematicians on these forums who know about the subject.

In the set of all prime numbers, has the integer gaps between two prime numbers been studied? I mean, do mathematicans know what the largest difference is between two prime numbers?

Im interested in this subject and I would like to know..

Thanks in advance.

--rad
 
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  • #2
The gap can be arbitrarily large. Just consider n!+2, n!+3,...n!+n.

Lots of work has been done, you might want to take a look at Caldwell's prime pages: http://primes.utm.edu/notes/gaps.html
 
  • #3
Since the primes have measure 0, prime gaps must be unbounded in length. Think about it -- if every k integers had a prime number for some fixed k, then some primes would have common (nontrivial) factors.
 
  • #4
i refer you to the work of helmut maier.

Helmut Maier
Primes in short intervals.

Source: Michigan Math. J. 32, iss. 2 (1985), 221
 
  • #5
CRGreathouse said:
Since the primes have measure 0

the integers have measure zero, but the gaps between consecutive integers is not unbounded.
 
  • #6
matt grime said:
the integers have measure zero, but the gaps between consecutive integers is not unbounded.

What I mean is that for a set [tex]X\in\mathbb{N}[/tex] with

[tex]\lim_{n\rightarrow\infty}\frac1n\sum_{x\in X|x\le n}x=0[/tex]

[tex]\forall n\in\mathbb{N}\;\;\exists m\in\mathbb{N}[/tex] such that there is no [tex]x\in X[/tex] with [tex]m\le x\le m+n[/tex]. (The set of primes is of course such a set by the PNT.) I'm sorry if I was ambiguous.
 
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  • #7
You want:

[tex]\lim_{n\rightarrow\infty}\frac1n\sum_{x\in X|x\le n}1=0[/tex]

or equivalently here pi(n)/n->0 as n->infinity. In otherwords, the asymptotic density of the primes is zero.
 
  • #8
According to Bertrand's postulate, there is at least one prime between n and 2n-2, for any n>3. I wonder if there is a theorem about the number of primes between n and 2n exclusive (see http://www.research.att.com/~njas/sequences/A060715 [Broken] ), because that number seems to be steadily increasing over a sufficiently large period of the sequence (sorry if this is not precise enough); what I mean is that, for n=5, for instance, the number of primes between n and 2n is 1, but, it seems, for any n > 5 the number of primes between n and 2n is greater than 1; similarly, for n= 8, the number of primes between n and 2n is 2, but for any n>8, the number of primes between n and 2n is greater than 2 (?), and so on.

If it is true that for any m >= 1 there is an n for which the number of primes between k and 2k, k>=n, is greater than m (is it?) then there is a limit on prime gaps as well, depending on m (or n), I think (although Bertrand's postulate itself puts a limit on prime gaps, depending on n).

What I mean by Bertrand's postulate putting a limit on prime gaps is that for any prime p, there is another prime between p+1 and 2p.
 
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  • #9
pi(2n)-pi(n)~n/log(n) by the prime number theorem, so the number of primes in [n,2n] tends to infinity as n does.

The bound Bertrands puts on the gap to the next prime is pretty far from what's known to be true (though correspondingly simpler to prove!). For example, if n is large enough, we can guarantee a prime in [n,n+n^0.525].
 
  • #10
CRGreathouse said:
What I mean is that for a set [tex]X\in\mathbb{N}[/tex] with

[tex]\lim_{n\rightarrow\infty}\frac1n\sum_{x\in X|x\le n}x=0[/tex]

[tex]\forall n\in\mathbb{N}\;\;\exists m\in\mathbb{N}[/tex] such that there is no [tex]x\in X[/tex] with [tex]m\le x\le m+n[/tex]. (The set of primes is of course such a set by the PNT.) I'm sorry if I was ambiguous.

density, not measure.
 
  • #11
shmoe said:
You want:

[tex]\lim_{n\rightarrow\infty}\frac1n\sum_{x\in X|x\le n}1=0[/tex]

or equivalently here pi(n)/n->0 as n->infinity. In otherwords, the asymptotic density of the primes is zero.

Oops, you're absolutely right. That's what I meant.
 

1. What are prime number gaps?

Prime number gaps refer to the difference between two consecutive prime numbers. For example, the prime number gap between 5 and 7 is 2, between 11 and 13 is 2, and between 17 and 19 is also 2. This gap can vary in size and is an important concept in number theory.

2. What is the largest known prime number gap?

The largest known prime number gap is the gap between the 9,991,993rd and 9,991,995th prime numbers, which is 2. This was discovered in 2016 by mathematicians at the University of Central Missouri and University of Tennessee.

3. Is there a pattern to prime number gaps?

There is no known pattern to prime number gaps. While some mathematicians have attempted to find patterns, there is currently no consensus on a formula or pattern that can predict the prime number gaps. This makes prime number gaps a fascinating and challenging topic in mathematics.

4. Can prime number gaps be negative?

No, prime number gaps cannot be negative. This is because prime numbers are positive integers and therefore the difference between two prime numbers will always be a positive integer. Negative numbers are not considered prime numbers.

5. Why are prime number gaps important?

Prime number gaps are important because they provide insight into the distribution of prime numbers. The gaps between consecutive prime numbers can vary greatly, but there are patterns that can be observed. Understanding these patterns can help mathematicians make progress in solving the unsolved problems related to prime numbers, such as the Twin Prime Conjecture and the Goldbach Conjecture.

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