A major subject of interest among mathematicians who work in number theory is the study of prime numbers. Prime numbers are divisible only by 1 and themselves, and large primes are used, among other things, in encrypting data. But perhaps, the best motivation for mathematicians to study them has been the aesthetics of seeing patterns among them. A new and interesting pattern has emerged among prime numbers thanks to the work of Kannan Soundararajan and Robert Lemke Oliver of Stanford University. They have found a pattern in the last digits of successive prime numbers, by analysing numerically the first 100 million numbers. For example, they see that a prime number ending in 9 is much less likely to be followed by another prime number ending in 9; it is more likely to be followed by one ending in 1, and so on. This study is important because it is a hitherto unexplored idea in the quest of understanding the blend of random distribution and patterns in prime numbers.
Though prime numbers occupy definite positions on the number line, thereby are not random at all, studying distributions of numbers comes in useful to predict properties which haven’t been proved yet. One example is the twin primes conjecture which can be simply stated that for infinitely many prime numbers, n, the number n +2 is also a prime.
“Randomness enters in as a convenient model to help understand which properties are likely to happen infinitely often and which we should expect to be so rare that they may only occur finitely many times. The guiding principle is that once we take into account all the facts that we are aware of, what remains may be expected to be roughly random,” says Kannan Soundararajan in an email to this correspondent.
For example, this mixture of randomness and order is taken into account by the Hardy Littlewood conjecture, from which Soundararajan and Lemke Oliver’s results take off: Says Dr Soundararajan, “The Hardy-Littlewood conjecture takes into account… facts of [numbers] being odd or even, or being a multiple of 3 or 5 or 7 etc, and after taking all these known facts into account pretends that the rest is random. This precise conjecture was formulated almost a hundred years back and has been extensively tested, although there is as yet no mathematical proof.”
Inspiration for their work came at a talk by Tadashi Tokieda on the “rock, paper, scissors in probability” “We started thinking about this inspired by a lecture of a colleague Tadashi Tokieda on curious non-transitive phenomena (like the rock-paper-scissors game) in coin tosses. That made me wonder if there were similar phenomena in the primes, which eventually led to this joint work with my colleague Robert Lemke Oliver,” Dr Soundararajan says.
Kannan Soundararajan is a leading mathematician who also shared the first SASTRA Ramanujan prize with Manjul Bhargava. This is a prize given for outstanding work done by mathematicians under 32 years of age.
