# You're reading: Posts Tagged: prime numbers

### Bound on prime gaps bound decreasing by leaps and bounds

Update 17/06/2013: The gap is down to 60,744. That’s a whole order of magnitude down from where it started!

When Yitang Zhang unexpectedly announced a proof that that there are infinitely many pairs of primes less than 70 million apart from each other – a step on the way to the twin primes conjecture – certain internet wags amused themselves and a minority of others with the question, “is it a bigger jump from infinity to 70 million, or from 70 million to 2?”.

Of course the answer is that it’s a really short distance from 70 million to 2, and here’s my evidence: the bound of 70 million has in just over three weeks been reduced to just a shade over 100,000.

### “Bounded gaps between primes” by Yitang Zhang now available

To complete the story started as a rumour report in ‘Primes gotta stick together‘ and confirmed in ‘Primes really do stick together‘, here we report that Annals of Mathematics has posted the PDF of ‘Bounded gaps between primes‘ by Yitang Zhang on its ‘to appear in forthcoming issues’ page. After the seminar on 13th May, Zhang apparently submitted a revised manuscript on 16 May, which was accepted 21 May 2013. So if you’ve been itching for details, here’s your chance (assuming you have access to a subscription to Annals).

Here’s the abstract:

It is proved that $\liminf_{n\to \infty}\, (p_{n+1} – p_n) < 7 \times 10^7 \text{,}$ where $p_n$ is the $n$-th prime.

Our method is a refinement of the recent work of Goldston, Pintz and Yildirim on the small gaps between consecutive primes. A major ingredient of the proof is a stronger version of the Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime divisors only (see Theorem 2), but it is adequate for our purpose.

The paper: Bounded gaps between primes by Yitang Zhang, in Annals of Mathematics.

### Primes gotta stick together

Update 14/05/2013: The seminar was successful: Zhang announced that his proof has already been refereed for the Annals, and everyone seems happy with it.

Hard Maths news now: there’s a rumour going round that Yitang (Tom) Zhang of the University of New Hampshire reckons he can prove that there are infinitely many different pairs of primes at most 70,000,000 apart.

### More experimental evidence for the infinitude of the primes

In a classic example of the intersection between maths and news, there’s been a new Mersenne prime discovered! Mersenne primes are numbers of the form $2^p – 1$, where $p$ is a prime number. They’re highly valued as a source of large prime numbers, since testing the primality of a (suspected) prime of this form is much easier than for general prime numbers.

### Factor Conga

Quite a few designery visualisations of the prime numbers have been put out on the web recently, to varying degrees of success. Most of the time they look pretty but don’t tell you very much; the most recent example I can think of is El Patrón de los Números Primos by Jason Davies.

A few weeks ago Brent Yorgey posted on his excellent blog The Math Less Traveled some really nice “factorization diagrams“, along with the code to produce them. Straight away, anyone with a text editor and a knack for fancy web coding set to work making the animated version that was so clearly required.

Stephen von Worley has made, I think, the nicest one. He calls it the Factor Conga. Sit back and enjoy the mysteries of the natural numbers as they dance their beguiling dance!

### Interesting Esoterica Summation, volume 4

Dust off your thinking hat and do some mind-stretches because here’s another course of Interesting Maths Esoterica! It’s been several months since the last volume so this is quite a big post. I won’t mind if you skim it.

In case you’re new to this: every now and then I encounter a paper or a book or an article that grabs my interest but isn’t directly useful for anything. It might be about some niche sub-sub-subtopic I’ve never heard of, or it might talk about something old from a new angle, or it might just have a funny title. I put these things in my Interesting Esoterica collection on Mendeley. And then when I’ve gathered up enough, I collect them here.