Patterns in Prime Numbers?

In March 2016, New Scientist reported on mathematicians finding a ‘pattern’ in prime numbers. For example, a prime ending in 1 was only followed by another prime ending in 1 18.5% of the time, rather than 25%, as would be expected if they were truly random.

I conducted my own analysis on the first 1952 primes.

Puzzling Primes

By Joel Johnson

Mathematicians have long assumed that primes are completely random – therefore the chance of a prime ending in a 1 being followed by another prime ending in a 1 should be 25%. This is all primes greater than 5 end in a 1, 3, 7 or 9.

However, this assumption has recently been shaken by the discovery of unexpected trends in the last number of primes. In March, New Scientist reported that Stanford University researchers had discovered that primes ending in a 1 have only a 18.5% chance of being followed by another prime ending in a 1 (1).

Here, the first 9592 primes (up to 100,000) were analysed. It was found that for a prime ending in 1, the chance of the following prime also ending in a 1 were 14.7%. It was been reported that the last digit pattern began to wear off as the primes tended toward infinity (1), which explains why the chance of a 1 following a 1 was much lower in this study.
The chance of a 3 being followed by a 3 was 12.8%, while the chance of a 7 being followed by a 7 was 12.9%. The probability of a 9 following a 9 was higher, at 15.3%. The full data are shown in the graphs below.

primes-overall-patternprimes-1-followed-byprimes-3-followed-byprimes-7-followed-byprimes-9-followed-by

References:
1. Aron, J 2016, ‘‘Random’ primes pair up on the sly’, New Scientist, iss 3065, 19 March 2016, p. 12.

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