Pattern discovered in prime numbers

Prime numbers turn out to be less random than thought and even show a pattern. This was discovered by two mathematicians from Stanford University in the United States. It turns out that prime numbers that follow each other are less likely to have the same number at the end.

The latter means that a prime number ending in 1 is less likely to be followed by another prime number ending in 1. If primes were random, this would not be the case. Now prime numbers are not random either, but they behave that way in many ways. Scientists consider the successive sequence of primes to be pseudorandom because there is no structure to indicate where a prime number occurs in the sequence. However, the researchers Kannan Soundararajan and Robert Lemke Oliver discovered an anomaly in the randomness, initially at the number 1. Prime numbers after 2 and 5 always end in 1, 3, 7 or 9. A prime number is a natural number and can only be divided by 1 and himself.

The researchers found that in the first 100 million primes, a prime number ending in 1 was followed by another prime number ending in 1 in only 18.5 percent of the cases. If prime numbers were really random, the next number would be in 25 percent of the cases. should end in 1. Prime numbers with a 9 at the end followed in 22 percent of the cases a prime number with a 1 at the end. Prime numbers ending with a 7 or 3 occur at 30 percent each.

The same pattern appeared to apply to prime numbers ending in 3, 7 and 9, which were also followed least often by a prime number ending in the same digit. Despite the fact that the pattern weakens with higher prime numbers – the researchers checked numbers up to a few trillion – the deviation remained visible.

Soundararajan got the idea to investigate this after a lecture on tossing. The lecture stated that if Alice tosses a coin until she sees a head followed by a tail, and Bob toss a coin until he sees two heads in succession, Alice will have to toss an average of four times, compared to Bob’s six times.

Soundararajan wondered if this strange phenomenon also occurs in other fields. Because he has been working with prime numbers for years, he decided to see if the same is going on here. That turned out to be the case. He looked at prime numbers with base 3, where roughly half of the prime numbers end in 1 and half end in 2. Prime numbers below 1000 in base 3 that ended in 1 were followed more than twice as many by a prime number ending in 2, and vice versa.

After Soundararajan showed his findings to Oliver, he wrote a program to search much further along the line of primes; namely by the first 400 billion primes. That showed the same. This also turned out to be the case not only for base 3 and 10, but also for other bases.

Why the last digit of a prime number does not appear to be randomly distributed is not entirely clear. The researchers suspect it has to do with how often pairs, groups of three, and larger groups of prime numbers occur, as predicted by the k-tuple conjecture.

According to the researchers, their finding does not seem to affect the practical use of prime numbers, such as for cryptography.

The paper can be found on the arXiv server.