## Monday, March 14, 2016

### Prime Jive

Mathematicians since Euclid have known about prime numbers, and there's evidence that the ancient Egyptians might have had some understanding of them as well.

They're the whole numbers larger than 1 that have only 1 and themselves as divisors. So 2 is one, as is 3. But 4 isn't, because you can also divide 4 by 2. The problem is that even though they're easily defined, they're not so easily found. They occur totally at random, and no equation or formula has ever been discovered that will predict prime numbers. Large groups of numbers can be ruled out, of course. After 2, there are no more even primes. And after 5, there are none that end in 5.

But that still leaves everything up in the air if number ends in a 1, 3, 7 or 9, until whatever number in question is subjected to some formula to determine if it is a prime or composite number. Before computers, division was one way to test a prime -- try to divide it by every number less than it was, until you either found or didn't find another divisor. This takes a great deal of time, and even computers don't use that technique now as the prime search moves into numbers so large they have no existence in the material universe. There aren't enough of anything that you could use the numbers under consideration to count the set.

Since the sequence of whole numbers is infinite -- for every number n, there will always be a number n+1 -- there are an infinite number of primes. Several theorems prove this, ranging from Euclid himself to modern proofs dealing with mathematical topology. The other thing about primes is that they were always believed to be completely random. That, in fact, was part of the problem in trying to predict them. They followed no pattern.

Or did they? Kannan Soundararajan and Robert Lemke Oliver of Stanford University discovered that prime numbers may not be completely predictable, but they do seem to have some "habits," so to speak. The pair studied the first billion prime numbers and found that if  a prime number ends in 9, for example, the next prime number is 65 percent more likely to end in a 1 than in another 9. Similar statistical clumpings show up for 3 and 7 as well. Prime numbers don't seem to "like" being followed by primes that end in the same digit.

Soundararajan and Oliver show that primes are still random. A prime ending in 3 may be more likely to be followed by one ending in 1, 7 or 9, but there's still a chance the next one ends in 3. And even if it doesn't, the other ending digits offer plenty of choices. But, they point out, what they have found means that some randoms are more orderly than others. And the real result of this finding may be that it spurs some more research into these curiously significant oddball numbers. “You could wonder, what else have we missed about the primes?” Montreal number theorist Andrew Granville asked in the story at Quanta magazine. It seems that sometimes the best-known things might not be known so well after all.