When Soundararajan first told Maynard what the pair had discovered, “I only half believed him,” Maynard said. “It’s the rate at which they even out which is surprising to me,” said James Maynard, a number theorist at the University of Oxford. The biases that they found appear to even out, little by little, as you go farther along the number line - but they do so at a snail’s pace. Lemke Oliver and Soundararajan discovered that this sort of bias in the final digits of consecutive primes holds not just in base 3, but also in base 10 and several other bases they conjecture that it’s true in every base. The primes “really hate to repeat themselves,” Lemke Oliver said. Lemke Oliver again found that primes seem to avoid being followed by another prime with the same final digit. He immediately wrote a program that searched much farther out along the number line - through the first 400 billion primes. Soundararajan showed his findings to postdoctoral researcher Lemke Oliver, who was shocked. Likewise, a prime ending in 2 prefers to be followed a prime ending in 1. Looking at prime numbers written in base 3 - in which roughly half the primes end in 1 and half end in 2 - he found that among primes smaller than 1,000, a prime ending in 1 is more than twice as likely to be followed by a prime ending in 2 than by another prime ending in 1. Since he has studied the primes for decades, he turned to them - and found something even stranger than he had bargained for. Soundararajan wondered if similarly strange phenomena appear in other contexts. 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. “That’s what we think we’ve done.” Prime Preferences “Can we redefine what ‘random’ means in this context so that once again, looks like it might be random?” Soundararajan said. Yet the pair’s work doesn’t upend the notion that primes behave randomly so much as point to how subtle their particular mix of randomness and order is. Even after having seen Lemke Oliver and Soundararajan’s analysis of their phenomenon, he said, “it still seems like a strange thing.” “I can’t believe anyone in the world would have guessed this,” Granville said. Most mathematicians would have assumed, Granville and Ono agreed, that a prime should have an equal chance of being followed by a prime ending in 1, 3, 7 or 9 (the four possible endings for all prime numbers except 2 and 5). This conspiracy among prime numbers seems, at first glance, to violate a longstanding assumption in number theory: that prime numbers behave much like random numbers. I thought, ‘For sure, your program’s not working.’” When he first heard the news, he said, “I was floored.

The discovery is the exact opposite of what most mathematicians would have predicted, said Ken Ono, a number theorist at Emory University in Atlanta. “We’ve been studying primes for a long time, and no one spotted this before,” said Andrew Granville, a number theorist at the University of Montreal and University College London.

Conspire 2016