Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them.
Profile: Breaking down the problem of bound gaps [New Yorker]: After graduating with a Ph.D. in algebraic geometry from Purdue in 1991, Yiting Zhang kept the books for a friend's Subway franchise and found other odd jobs before taking up a part-time calculus teaching position at the University of New Hampshire in 1999.
“For years, I didn’t really keep up my dream in mathematics,” he said.He published one paper in 2001. Then, in 2013, he submitted "Bounded Gaps Between Primes" to Annals of Mathematics, one of the most prestigious journals in the field, which contained a proof for a finite bound within which there exist an infinite number of pairs of primes. It was a stunning mathematical breakthrough. [more inside]
“You must have been unhappy.”
He shrugged. “My life is not always easy,” he said.
It's a bit late for the holiday, but math(s) comedian Helen Arney sings about her Christmas wish -- the largest known Mersenne Prime, Mersenne 48. [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]
If the NSA is able to break through banks' computer security, does that mean it solved the prime factorization problem? The New York Times reported recently that “the agency has circumvented or cracked much of the encryption, or digital scrambling, that guards global commerce and banking systems.” Since banks' encryption codes rely on the fact that nobody knows how to find the prime factors of really large numbers, it could mean that the NSA has found a way to do that. Or it could mean that the NSA has simply gotten lots of banks to give up their information, or found other ways around their encryption. But if they've cracked this long-standing math problem, might the secret leak? What would be the effects?
This afternoon, Yitang Zhang of the University of New Hampshire gave a special seminar at Harvard, in which he announced that he had proved that there are infinitely many pairs of prime numbers separated by no more than 70,000,000, a result differing only by a constant factor from the venerable twin prime conjecture. Dan Goldston, who together with Yildirim and Pintz made the last major advance on prime gaps, said, ""I was doubtful I would ever live to see this result." Not enough excitement for one day? Harald Helfgott has just posted to the arXiv a proof of the ternary Goldbach conjecture: every odd number is the sum of three primes.
"Each prime number is represented by a bright, white square, whereas a non-prime ("composite") is grey. Visitors can select difference spatial arrangements of these numbers, ranging from several variants of the well-known Ulam Spiral, over the Archimedian spiral, to the more sophisticated 3D Hilbert curves." [more inside]
What is the smallest prime? "It seems that the number two should be the obvious answer, and today it is, but it was not always so. There were times when and mathematicians for whom the numbers one and three were acceptable answers. To find the first prime, we must also know what the first positive integer is. Surprisingly, with the definitions used at various times throughout history, one was often not the first positive integer (some started with two, and a few with three). In this article, we survey the history of the primality of one, from the ancient Greeks to modern times. We will discuss some of the reasons definitions changed, and provide several examples. We will also discuss the last significant mathematicians to list the number one as prime."
The Prime Game is not really much of a game, but it is a neat & little-known fact about the decimal representation of prime numbers.
The math geeks have done it again. Yet another prime number which, when converted to binary, contains DeCSS: this one's an x86 Linux ELF executable. Only took a weekend of hacking to do, and it's only 752 bytes! You know what they say: when prime numbers are outlawed, only outlaws will have prime numbers.
Mersenne Prime Search is a distributed computing project much like Seti@home, except instead of searching for aliens, you're in the running for $100,000 and a place in math history (shouldn't your computer actually be the one that goes into the math history books?).