Prime numbers - the unique numbers that can only be divided by one and themselves - are used by mathematicians as tools for various purposes, including discovering others in nature and helping them uncover larger primes. Now, a team of researchers has revealed a strange new property about them that conflicts with the previous understanding that they occur in a completely random pattern across integers.
The new finding is that prime numbers seem to repel other potential prime numbers that end in the same digit, a pattern that isn't seen in the prime number themselves, but in the last digit of the prime number that comes after them.
"We've been studying primes for a long time, and no one spotted this before," said Andrew Granville, a number theorist who wasn't involved in the study. "It's crazy."
Why is this such a big deal? In the past, mathematicians have suggested that over a large enough sample of numbers, prime numbers occur randomly, meaning that they shouldn't be influenced by prime numbers that come before or after. However, the new study suggests otherwise.
Stanford University researchers Kannan Soundararajan and Robert Lemke Oliver decided to perform a randomness check on the first 100 million prime numbers and found that prime numbers ending in one were followed by another prime number ending in one just 18.5 percent of the time. This is a big difference from the 25 percent expected if prime numbers truly do have a random distribution.
In addition, they found that the chance of a prime number ending in one being followed by a prime number ending in three or seven was 30 percent, but for nine, the chance was just 22 percent, supporting the theory that prime numbers don't like to repeat themselves.
Soundararajan and Lemke Oliver believe that the reason for these findings lies in the idea called the k-tuple conjecture. This idea deals with how larger sets of prime numbers are distributed, and the team believes that it is the force behind the new biases that they have discovered.
Further research will need to be conducted in order to determine if this new property is an isolated pattern or something that influences other mathematical structures. Nevertheless, the team believes that other mathematicians should look for similar patterns in related contexts, such as prime polynomials.
The findings were published March 11 on the pre-press site arXiv.
