Univalent Foundations Redefines Mathematics - "When a legendary mathematician found a mistake in his own work, he embarked on a computer-aided quest to eliminate human error. To succeed, he has to rewrite the century-old rules underlying all of mathematics." (previously) [more inside]
Understanding e to the pi i - "An intuitive explanation as to why e to the pi i equals -1 without a hint of calculus. This is not your usual Taylor series nonsense." (via via; reddit; previously) [more inside]
Closing in on the twin prime conjecture (Quanta) - "Just months after Zhang announced his result, Maynard has presented an independent proof that pushes the gap down to 600. A new Polymath project is in the planning stages, to try to combine the collaboration's techniques with Maynard's approach to push this bound even lower." [more inside]
In August of last year, mathematician Shinichi Mochizuki reported that he had solved one of the great puzzles of number theory: the ABC conjecture (previously on Metafilter). Almost a year later, no one else knows whether he has succeeded. No one can understand his proof.
Using computer systems for doing mathematical proofs - "With the proliferation of computer-assisted proofs that are all but impossible to check by hand, Hales thinks computers must become the judge." [more inside]
FatFonts creates numerical fonts where the amount of ink/pixels for each number is in direct proportion to its value.
New math theories reveal the nature of numbers [1,2] - "We prove that partition numbers are 'fractal' for every prime. These numbers, in a way we make precise, are self-similar in a shocking way. Our 'zooming' procedure resolves several open conjectures, and it will change how mathematicians study partitions." (/.|via) [more inside]
In September 2006 the largest known prime number, a 9.8 million digit number, was discovered. If you find one over ten million digits you can win US$100,000 (of which you get to keep $50,000). No maths is required - just download the software and you're away. Warning: it takes about a month to run one primality check so some patience is required. Look out though Cooper and Boone look like they might beat you to it.