Exploring Bertrand's Postulate: Maximal Prime Gaps and Conjectures

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In summary: This is interesting. I wonder why nobody thought that the Frey curve might be used to prove the FLT before?
  • #1
huba
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I hope I am correct in saying that Bertrand's postulate can be rephrased this way: the maximum prime gap following prime p is p-3, if p > 3.

Is this the closest proven result for the maximal prime gap?
Wolfram MathWorld mentions 803 as a large known prime gap following 90874329411493, and 4247 following 10^314-1929 !
 
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  • #3
huba said:
I hope I am correct in saying that Bertrand's postulate can be rephrased this way: the maximum prime gap following prime p is p-3, if p > 3.

Is this the closest proven result for the maximal prime gap?

Of course that is not even close to the expected maximal prime gap, which should be on the order of [itex](\log p)^2.[/itex]

Using the bounds of Dusart (along with the classical bound of Rosser) it should not be hard to show that [itex]p_{n+1}-p_n<2n+(1+\varepsilon)\log n[/itex] for some value like [itex]\varepsilon=0.1[/itex].* This is [itex]O(n/\log n),[/itex] so only slightly better than the bounds from Bertrand & Tchebychev.

There are asymptotic results that are far better, showing [itex]p_{n+1}-p_n<p_n^{1/2+\eta}[/tex] for small rational [itex]\eta[/itex] and sufficiently large n.

* If I have not miscalculated, the bound is [tex]2n+\log n+(1+\varepsilon_0)\log\log n+\varepsilon_1n+\varepsilon_2[/tex] for various [tex]\varepsilon\to0[/tex]. These are values like [tex]\log(n+1)-\log(n),[/tex] for example.
 
  • #4
Thanks.
Why are the two prime gap lengths, 803 and 4247, quoted from Wolfram MathWorld (under "Prime Gaps"), odd numbers?
 
  • #5
Is Goldbach's conjecture difficult to prove because there isn't a tight enough upper bound for prime gaps?
Is it possible to construct a sequence a(n) of odd numbers, with a gap, depending on, and increasing with, n, between a(n+1) and a(n), such that all even integers greater than a given number can be written as the sum of two terms of a(n)? If it could be shown that a(n+1) - a(n) >= the maximal prime gap following the nth prime, would that prove the conjecture?
 
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  • #6
huba said:
Thanks.
Why are the two prime gap lengths, 803 and 4247, quoted from Wolfram MathWorld (under "Prime Gaps"), odd numbers?

Their definition of a prime gap differs from the more common ("standard") one by 1.
 
  • #7
huba said:
Is Goldbach's conjecture difficult to prove because there isn't a tight enough upper bound for prime gaps?
Is it possible to construct a sequence a(n) of odd numbers, with a gap, depending on, and increasing with, n, between a(n+1) and a(n), such that all even integers greater than a given number can be written as the sum of two terms of a(n)? If it could be shown that a(n+1) - a(n) >= the maximal prime gap following the nth prime, would that prove the conjecture?

It's hard to prove because nontrivial additive facts about the primes are hard to come by. Primes 'work' multiplicatively, so addition is a bit unnatural.
 
  • #8
Goldbach conjecture and prime gaps

huba said:
Is Goldbach's conjecture difficult to prove because there isn't a tight enough upper bound for prime gaps?
Is it possible to construct a sequence a(n) of odd numbers, with a gap, depending on, and increasing with, n, between a(n+1) and a(n), such that all even integers greater than a given number can be written as the sum of two terms of a(n)? If it could be shown that a(n+1) - a(n) >= the maximal prime gap following the nth prime, would that prove the conjecture?

Well, that's possible because anything's possible, but it doesn't seem very likely to me anyway. If there are heuristic arguments indicating that this might be the case, I would like to know about them.

Page 35 of Manin and Panchishkin indicates gives heuristic arguments indicating that Goldbach's conjecture might follow from the application of the Hardy-Littlewood circle method developed by Hardy and Littlewood and applied by Hardy, Littlewood and Ramanujan to obtain the Hardy-Ramanujan asymptoic formula for the partition function. As far as I know, that is the current "conventional wisdom" in the field.

However, recent events insert a large degree of uncertainity.

Namely, when Frey presented his now-famous curve with a heuristic argument that if the modularity conjecture could be proven then FLT would follow, the conventional wisdom was that the signifiance of the Frey curve (so-called because although many people had studied and published results on that curve before Frey - as far back as the early 20th century, none said anything that suggested any possibility of using the Frey curve to prove the FLT, as far as I know) was that we were generations away from proving the modularity conjecture.

Wiles stood up at one of the meetings where Frey presented his curve and expressed the opposite opinion; namely that the modularity conjecture would be the path to the (first) proof of FLT. Everybody in the room disagreed with Wiles opinion.

However, when Ribet and Serre's formal proof of Frey's heuristic claims came out, the prevailing opinion among algebraic geometric number theorists changed to consider Wiles opinion as a real option. Wiles, of course, proved he was (almost) right in 1993, and finished the job as kind of a group effort but especially with Taylor in 1994.

A similar thing happened a few years later with the Poincare conjecture.

So, my guess is that the "prevailing wisdom" now is that "anything is possible."

Deacon John
 

1. What are prime gaps?

Prime gaps are the difference between two consecutive prime numbers. For example, the gap between 5 and 7 is 2, and the gap between 11 and 13 is 2 as well.

2. How are prime gaps calculated?

To calculate prime gaps, you simply subtract the smaller prime number from the larger one. This will give you the difference, or gap, between the two numbers.

3. How do prime gaps relate to the distribution of prime numbers?

The study of prime gaps is closely related to the distribution of prime numbers. In fact, the famous twin prime conjecture, which states that there are infinitely many pairs of prime numbers with a gap of 2 between them, is still an unsolved problem in mathematics.

4. Are there any patterns or trends in prime gaps?

While there are some observed patterns in prime gaps, such as the tendency for smaller gaps to occur more frequently than larger ones, there is no known formula or algorithm for predicting prime gaps. They are considered to be random and unpredictable.

5. How are prime gaps used in cryptography?

Prime gaps are important in cryptography because they are used to generate large and secure prime numbers, which are essential for many encryption algorithms. The randomness and unpredictability of prime gaps helps to ensure the security of these systems.

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