Two mathematicians from Stanford University, Kannan Soundararajan and Robert Lemke Oliver (pictured) discovered a previously unknown property of prime numbers. They found that the chances of a prime ending in 9 being followed by a number ending in 1 are 65% greater than the chances of being followed by a number ending in 9 again. This assumption was numerically tested by computer methods for billions of known primes.
According to Ken Ono, a mathematician at Emory University in Atlanta, this assumption is essentially contrary to the expectations of most mathematicians. Previously, it was believed that prime numbers for the most part behave quite randomly. Most theorists would agree on the assumption that the odds of having one of the possible digits for prime numbers (1, 3, 7, 9) at the end are approximately equal for all such numbers.
Andrew Granville of the University of Montreal stated that “we have been studying prime numbers for a very long time and no one noticed it before. This is some kind of madness. I can't believe anyone could think of this. It looks very strange."
Soundarajan said that he was inspired by a lecture by the Japanese mathematician Tadashi Tokieda that gave him the idea of testing for "randomness" in the world of prime numbers. In it, he gave an example from the theory of probability. If Alice flips coins until she gets tails following heads, and Bob flips two heads in a row, then Alice will need four coin tosses on average, while Bob will need six. In this case, the probability of getting heads and tails is the same.
Since Soundarajan was interested in prime numbers, he turned to them in search of hitherto unknown distributions. He found that if you write the primes in the ternary system, in which about half of the primes end in 1 and half in the number 2, then for primes less than 1000, after the number ending in 1, it is twice as likely follow a number ending in 2 than 1 again.
He shared an interesting discovery with another scientist, Lemke Oliver, and he, amazed at this fact, wrote a program that checked how things are with the distribution of numbers in the first 400 billion primes. The results confirmed the hypothesis - as Oliver put it, prime numbers "hate repetitions." The assumption was tested for both decimal notation and some other number systems.
It is not yet known whether this property is some kind of separate phenomenon, or is associated with deeper properties of prime numbers that have not been discovered so far. As Granville said, "I wonder what else could we have missed in the prime numbers?"