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“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.
Integer sequence review: A000959
The Online Encyclopedia of Integer Sequences contains over 200,000 sequences. It contains classics, curios, thousands of derivatives entered purely for completeness’s sake, short sequences whose completion would be a huge mathematical achievement, and some entries which are just downright silly.
For a lark, David and I have decided to review some of the Encyclopedia’s sequences. We’re rating sequences on four axes: Novelty, Aesthetics, Explicability and Completeness.
A000959
Lucky numbers.1, 3, 7, 9, 13, 15, 21, 25, 31, 33, 37, 43, 49, 51, 63, 67, 69, 73, 75, 79, 87, 93, 99, 105, 111, 115, 127, 129, 133, 135, 141, 151, 159, 163, 169, 171, 189, 193, 195, 201, 205, 211, 219, 223, 231, 235, 237, 241, 259, 261, 267, 273, 283, 285, 289, 297, 303, ...
First papers in Forum of Mathematics Pi and Sigma
I had hoped that The Future of Scholarly Mathematical Intercourse would arrive chaperoned by The Future of Publishing.
The first papers in Cambridge University Press’s new journals, Forum of Mathematics Pi and Forum of Mathematics Sigma, have been published — $p$-adic Hodge theory for rigid-analytic varieties by Peter Scholze in FoM Pi, and Generic mixing theory via vanishing Hodge models by Minhea Popa and Christian Schnell in FoM Sigma. But since the journals are more interesting for the medium they’re delivered by than their message, I’d like to take a look at the experience I had when accessing them.
EDF0 by Raven Kwok
[vimeo url=https://vimeo.com/43752422 w=600]
via NotCot.org
Integer sequence review: A005114
The Online Encyclopedia of Integer Sequences contains over 200,000 sequences. It contains classics, curios, thousands of derivatives entered purely for completeness’s sake, short sequences whose completion would be a huge mathematical achievement, and some entries which are just downright silly.
For a lark, David and I have decided to review some of the Encyclopedia’s sequences. We’re rating sequences on four axes: Novelty, Aesthetics, Explicability and Completeness.
Following last week’s palaver, we’re going to do our best to be serious this time. Game faces on. David promises there will actually be some maths in this sequence.
A005114
Untouchable numbers: impossible values for sum of aliquot parts of $n$.2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 206, 210, 216, 238, 246, 248, 262, 268, 276, 288, 290, 292, 304, 306, 322, 324, 326, 336, 342, 372, 406, 408, 426, 430, 448, 472, 474, 498, 516, 518, 520, 530, 540, 552, 556, 562, 576, 584, 612, 624, 626, 628, 658, ...
Axes to Axes
In which the intrepid maths-crime-fighting duo of Gale and Beveridge find themselves thrust back to a time before people could do maths properly.
It had been a quiet night at the Aperiodical police station. Apart from a few cases of broken scheduling in Excel formulas – nothing a bit of TIME() in the cells wouldn’t put right – there was nothing.
At 11pm, the phone rang. I looked at Sergeant Gale. Sergeant Gale pointedly looked at the phone, raised an eyebrow, and returned to his sudoku.
“Maths Police, bad graphs department. Constable Beveridge speaking, how can I help?”