The article may be a little heavy-going, but I know some of my readers will love it. It's not often number theory makes the headlines: Mathematicians Discover Prime Conspiracy. Maybe it's the idea of conspiracy—that always sells.
Two mathematicians have uncovered a simple, previously unnoticed property of prime numbers — those numbers that are divisible only by 1 and themselves. Prime numbers, it seems, have decided preferences about the final digits of the primes that immediately follow them.
Among the first billion prime numbers, for instance, a prime ending in 9 is almost 65 percent more likely to be followed by a prime ending in 1 than another prime ending in 9. In a paper posted online today, Kannan Soundararajan and Robert Lemke Oliver of Stanford University present both numerical and theoretical evidence that prime numbers repel other would-be primes that end in the same digit, and have varied predilections for being followed by primes ending in the other possible final digits.
The discovery is the exact opposite of what most mathematicians would have predicted... Most mathematicians would have assumed ... that a prime should have an equal chance of being followed by a prime ending in 1, 3, 7 or 9.
Soundararajan was drawn to study consecutive primes after hearing a lecture at Stanford by the mathematician Tadashi Tokieda, of the University of Cambridge, in which he mentioned a counterintuitive property of coin-tossing: If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses.
Soundararajan wondered if similarly strange phenomena appear in other contexts. Since he has studied the primes for decades, he turned to them — and found something even stranger than he had bargained for.
What does this mean for ordinary mortals? Who knows? It may mean nothing ... or it may lead to the next big break in cryptography. With math, anything's possible.
The infinite progression of numbers makes me think of God. :
In the coin-toss problem, the key is to look at the ways of getting the output you want in exactly 3 tosses. (There is one way for each output to win in exactly one toss.) For three tosses, the only way to get HH is THH, but you can get HT with HHT or THT. Note that this is not the same as saying that HT will probably appear before HH, since one of the ways we got HT started with HH.
