Following up on yesterday's words from Richard Elwes I'll toss out another bit from his wonderful volume "Math In 100 Key Breakthroughs" (pg. 386). We're all familiar with "twin primes" like 11 & 13, and even triplet primes like 3, 5, 7. One can also signify longer sequences of primes that are separated by equal gaps: 11, 17, 23, 29, for example have spacing 6-apart (there are other primes, 13, 19, interspersed, but we're ignoring them).
Richard Elwes asks, "How long can sequences like this be?" and replies further, "The search quickly becomes hard, as the individual numbers involved become very large, too. The longest currently known arithmetic progression of primes consists of 26, beginning with 43,142,746,595,714,191 and then increasing in steps of 544,680,710. It has long been conjectured that there should be arithmetic progressions of primes of every possible length. This idea dates back at least to 1770, to the work of Edward Waring and Joseph Louis LaGrange. But the conjecture resisted all attempts at proof until 2004, when Ben Green and Terence Tao collaborated to prove their stunning theorem.
"If you want a list of 100 primes, each exactly the same distance from the last, the Green-Tao theorem guarantees there will be such a list somewhere. It does not, however, provide much useful information about where to start looking!"
...mind… blown… yet… again. . . .
…and, as long as we're speaking about primes, I hope most of you saw Web cartoonist xkcd's recent effort on the Goldbach conjecture(s): http://xkcd.com/1310/
Finally, a safe, happy... and mathy NEW YEAR to one-and-all!!!