It's not every day that a brand new property is found in something as universal as prime numbers, but that's exactly what a pair of Stanford University mathematicians claim to have discovered. If verified, their finding would disprove a fundamental assumption long held about prime number behavior: that they occur randomly across the integers.
"We’ve been studying primes for a long time, and no one spotted this before," Andrew Granville, a number theorist at the University of Montreal who wasn't involved in the study, told Quanta magazine. "It’s crazy."
A prime number is, of course, a natural number greater than 1 that has no positive divisors other than 1 and itself. The key element that makes these numbers so fascinating to mathematicians is that they are essentially the building blocks from which the rest of the number line is constructed. Every other number can be created by multiplying primes together. So it seems that the key to understanding the fundamentals of arithmetic is to first unravel the mysteries of the primes.
One of these mysteries surrounds the unpredictable distribution of primes across the integers. There doesn't seem to be a pattern; the primes occur randomly. Or at least, that is what mathematicians have always believed.
It now turns out that assumption may be dead wrong.
Stanford researchers Kannan Soundararajan and Robert Lemke Oliver ran a program that essentially performed a randomness check on the first 100 million primes. To their surprise, a pattern was actually found, hiding beneath the figurative noses of mathematicians for ages. It turns out that prime numbers have a peculiar dislike for other would-be primes that end in the same digit.
For instance, it was found that a prime ending in 1 was followed by another prime ending in 1 only 18.5 percent of the time. Since all prime numbers after 5 can only end in one of four digits — 1, 3, 7 or 9 — that calculation should have been at 25 percent. That is, assuming the primes have a random distribution.
That wasn't the only weird result that the researchers found. It turns out that the chance of a prime ending in 1 being followed by a prime ending in 3 or 7 was roughly 30 percent, but for 9 it was only 22 percent. In other words, primes that end in 1 seem to have a bias for being followed by primes that end in 3 or 7. But why?
The researchers checked their work to make sure something else funky wasn't going on, but failed to discover any flaws. For instance, when the program was run using base systems other than 10 (such as is the case with our standard number system), the calculations still held. They even expanded their search to the first few trillion prime numbers, and couldn't shake the pattern.
So what's going on here? Luckily, Soundararajan and Lemke Oliver think they have an explanation. Their findings curiously fit with a well-established but unproven idea called the k-tuple conjecture. This conjecture essentially deals with how often pairs, triples and larger sets of primes will make an appearance, and how close together these sets should occur. It turns out that this system appears to force the primes to display the strong biases identified by the researchers.
Furthermore, it was found that as the primes stretch to infinity, the biases become increasingly less predictive. In other words, the larger the number of primes you look at, the more they tend toward the random distribution mathematicians are used to expecting. So perhaps the pattern eventually fizzles out completely.
Either way, this is certainly a discovery that is sure to rock the world of mathematics for many years to come.
“You could wonder, what else have we missed about the primes?” said Granville.
You can read more about the discovery in the science journal Nature.