Game of primes ends as mathematics gets too difficult - physics-math - 23 April 2014 - New Scientist

Game of primes ends as mathematics gets too difficult - physics-math - 23 April 2014 - New Scientist:
The twin prime conjecture posits that there are an infinite number of pairs of primes separated by two, such as 3 and 5. In May 2013, Yitang Zhang at the University of New Hampshire in Durham made the first major progress by proving that there are an infinite number of prime pairs separated by at most 70 million.

Since then, researchers have been advancing his methods to find prime pairs separated by smaller intervals...

In a few weeks they had reduced the gap to just under 5 million, and today it stands at 246. In theory, the methods used could find gaps as small as six, but the researchers say they have reached a point of diminishing returns.

Prime number enigma could be solved by simple networks - physics-math - 03 March 2014 - New Scientist

Prime number enigma could be solved by simple networks - physics-math - 03 March 2014 - New Scientist:

To investigate, Marián Boguñá of the University of Barcelona, Spain, and his colleagues turned the numbers from 2 to 1 billion into a network by linking composites to their prime building blocks. They also devised a way to generate similar networks using other simple rules to link the numbers, and wondered if they could get these networks to reproduce the pattern of links between primes and composites.

Their rules depend on probabilities, so the generated networks are different each time. On average, the researchers found that the webs are very close to the real network of primes and composites...

First Proof That Infinitely Many Prime Numbers Come in Pairs

First Proof That Infinitely Many Prime Numbers Come in Pairs: ...the gap between each prime and the next becomes larger and larger — on average. But exceptions exist: the ‘twin primes’, which are pairs of prime numbers that differ in value by 2. Examples of known twin primes are 3 and 5, or 17 and 19, or 2,003,663,613 × 2195,000 − 1 and 2,003,663,613 × 2195,000 + 1...

The new result, from Yitang Zhang of the University of New Hampshire in Durham, finds that there are infinitely many pairs of primes that are less than 70 million units apart without relying on unproven conjectures. Although 70 million seems like a very large number, the existence of any finite bound, no matter how large, means that that the gaps between consecutive numbers don’t keep growing forever...

Fiendish 'ABC proof' heralds new mathematical universe - physics-math - 10 September 2012 - New Scientist

Fiendish 'ABC proof' heralds new mathematical universe: Take 81 + 64 = 145, which breaks down into the prime building blocks 3 × 3 × 3 × 3 + 2 × 2 × 2 × 2 × 2 × 2 = 5 × 29. Simplified, the conjecture says that the large amount of smaller primes on the equation's left-hand side is always balanced by a small amount of larger primes on the right – the addition restricts the multiplication, and vice versa...
Rather than using sets, Mochizuki has figured out how to translate fundamental mathematical ideas into objects that only exist in new, conceptual universes. This allowed him to "deform" basic whole numbers and push their innate relationships – such as multiplication and addition – to the limit...

Logic blooms with new 11-set Venn diagram - physics-math - 10 August 2012 - New Scientist

Logic blooms with new 11-set Venn diagram: To find the rose-like diagram, the pair had to comb through myriad potential diagrams, represented as lists of numbers corresponding to the way the curves cross. Sifting through all of the possibilities for an 11-set diagram would be an impossible task even for the combined might of Earth's computers, so the researchers narrowed the options by restricting the search to diagrams with a property called crosscut symmetry, meaning that a segment of each set crosses all the other sets exactly once.

Freezing glass may shed light on a great mystery in mathematics

Freezing glass may shed light on a great mystery in mathematics: In glasses, atoms are freeze solid, arranged in a disorderly manner, while in crystals, they are arrayed in an orderly fashion. The way energy is distributed within disordered systems like glasses resembles a random landscape of hills and valleys. As the amount of energy within such a system is lowered, any travelers navigating this landscape would slow and eventually stop. The areas in which they would tend to freeze in place resemble the way numbers cluster with the Riemann zeta function.