Patterns discovered in primes

Mar

16

2016

Prime numbers are less random than expected and even show a pattern. Discovered two mathematicians at Stanford University in the United States. It appears to be that prime numbers which follow each other, less likely to have the same number at the end.

The latter means that a prime number on which the first ends less likely to be followed by a different prime number of prime numbers ending to 1. If would be arbitrary, this would not be the case. Now primes not random, but behave in such a way in many respects. Scientists consider the successive sequence of prime numbers as pseudo-random because not have to give them a structure is true in the series prevents a prime number. The researchers Kannan Soundararajan and Robert Lemke Oliver found out , however, an abnormality in the randomness, initially with the number 1. Primes end after 2 and 5 on 1, 3, 7 or 9. A prime number is divisible only by 1 and itself.

The researchers discovered that ending 1. in the first 100 million primes prime ended 1 was followed in only 18.5 percent of cases by another prime as prime numbers are really random, the next number would be in 25 percent of cases must end with one. Prime numbers with a 9 at the end followed in 22 percent of cases, a prime number having a 1 in the end on. Prime numbers ending with a 7 and 3 are each 30 percent.

A similar pattern was found for prime numbers ending in 3, 7 and 9 to apply: the least often were also followed by a prime number ending in the same digit. Despite the pattern is less pronounced at higher prime numbers – the researchers checked numbers to a few trillion – the uncertainty remained visible.

Soundararajan got the idea to investigate this after a lecture on toss. The lecture was held that if Alice coin tosses until a head by a coin followed behold, and Bob tosses a coin until two cups sees a row, Alice will have to toss an average of four times, at six times toss for Bob.

Soundararajan wondered if this strange phenomenon also occurs in other areas. Because he for years has been engaged in primes, decided he see if this same thing is going on. That was indeed the case. He looked at primes in base 3, in which roughly half of the primes in one and half ends 2. Prime numbers below 1000 in base 3, which ended up on 1 were more than twice as much followed by a prime number ending in two, and vice versa.

After Soundararajan showed his findings to Oliver, who wrote a program to search much further along the line of primes; namely through the first 400 billion primes. That showed the same to. This, too, is not only found to be the case 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, it is not entirely clear. The researchers suspect that it has to do with how often couples, prevent groups of three and larger groups of primes, as predicted by the k-tuple -vermoeden.

According to the researchers, their discovery seems to have no influence on practical use of prime numbers, such as cryptography.

The paper can be found on the arXiv server.

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