(This originally appeared over on my physics blog. Enjoy!)
Lately, as a form of review, I’ve been taking a quantum mechanics course on Coursera. (It was, in fact, that course which prompted me to recently post a derivation of the Schrödinger equation a few weeks ago.) A couple of the lectures were devoted to a brief introduction to Feynman’s path-integral formulation of quantum mechanics, something typically not brought up in courses at that level, which was a refreshing change of pace. A key component of deriving Feynman’s approach is Laplace’s method, a mathematical technique that I’ve probably not thought about since taking Mathematical Methods for Physicists way back in the Dark Ages when I rode a dinosaur to grad school. (Now, where the heck is my copy of Arfkin?) A review was definitely in order.
If you haven’t caught the Numberphile video series over on YouTube, you don’t know what you are missing. These short videos by Dr. James Grime and Brady Haran provide brief, simple-to-grasp explanations of a variety of somewhat sophisticated mathematical topics. For example, yesterday’s new video covered some territory to which I had not really given any thought in years: the fact that 0! is equal to 1.
“You’ve broken maths, Brady. Stop that!”
The Twin Prime Conjecture
First up, on April 17th (my birthday, no less), the journal Annals of Mathematics received a submission from Yitang Zhang of the University of New Hampshire purporting to at least partially prove the twin prime conjecture. The paper (available for preview here if you have access) will appear in an upcoming issue, but it is already creating a stir in the mathematics community. Building upon earlier work by Goldston, Pintz, and Yildrim (GPY), Zhang’s paper demonstrates that for any integer N less than 70 million, there are infinitely many pairs of primes that differ by N.
More at these links:
The Ternary Goldbach Conjecture
But wait, there’s more. H. A. Helfgott claims to a proven the ternary Goldbach conjecture.
In its most basic form, Goldbach’s conjecture states that every even integer greater than 2 can be expressed as the sum of two primes. The ternary Goldbach conjecture, also known as the weak or odd Goldbach conjecture, states that every odd number greater than 5 can be expressed as the sum of three primes (with repeats allowed). For example, 7 can be written as 2+2+3.
More at these links:
While I grind away at some content for this site, I hope you’ll enjoy this little exploration over at Scientific American of the relationship between the life cycle of cicadas and prime numbers….