Exploring Bertrand's Postulate: Maximal Prime Gaps and Conjectures

[huba]

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 !

 

[Santa1]

That is max prime gap after a prime p yes, I know an interesting article about a fairly new result on prime gaps:

http://www.ams.org/bull/2007-44-01/S0273-0979-06-01142-6/S0273-0979-06-01142-6.pdf

I don't know your background in number theory, but it has good introduction to the crámer model etc. If you atleast read the beginning you'll have some idea of the basic ideas.

 

[CRGreathouse]

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 (\log p)^2.

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

There are asymptotic results that are far better, showing p_{n+1}-p_n&lt;p_n^{1/2+\eta}[/tex] for small rational \eta and sufficiently large n.<br /> <br /> * If I have not miscalculated, the bound is 2n+\log n+(1+\varepsilon_0)\log\log n+\varepsilon_1n+\varepsilon_2 for various \varepsilon\to0. These are values like \log(n+1)-\log(n), for example.

 

[huba]

Thanks.
Why are the two prime gap lengths, 803 and 4247, quoted from Wolfram MathWorld (under "Prime Gaps"), odd numbers?

 

[huba]

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?

 

[CRGreathouse]

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.

 

[CRGreathouse]

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.

 

[DeaconJohn]

Goldbach conjecture and prime gaps

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