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Posted on 4 June by Brian Hayes Are the prime numbers sprinkled randomly along the number line, like windblown seeds? Primality is not a matter of chance but a product of simple arithmetic.
A number is prime if and only if no smaller positive integer other than 1 divides it evenly. The distribution of the primes looks random, with irregular gaps and clusters that seem quite haphazard.
As a matter of fact, the primes look random enough that you could play dice with them. Make a list of consecutive prime numbers perhaps starting with 11, 13, 17, 19. In other words, divide each prime by 7 and keep only the remainder. I have also simulated a million rolls of a six-sided die. Looking at the results of these Checkerboard wrapping paper exercises, can you tell which is which?
In both cases the numbers seem to be spread pretty evenly, without any obvious biases. The first table represents the prime residues mod 7. They have a somewhat flatter distribution than the simulated die, with smaller departures from the mean; the standard deviations of the two samples are 84 and respectively.
On the evidence of these tables it looks like either process could supply the randomness needed for a casual game of dice.
Individual events in the series must also be independent of one another. One roll of a die should have no effect on the outcome of the next roll.
As a test of independence, we can look at pairs of successive events. How many times is a 1 followed by another 1, by a 2, by a 3, and so on? If the process is truly random, all 36 pairs should be equally frequent, apart from small statistical fluctuations.
Here is the heatmap for the simulated fair die: Not much going on there. Almost all the counts are so close to the mean value that the matrix cells appear as a neutral gray; a few are very pale pink or blue.
Now for the corresponding matrix of consecutive primes mod 7: The heatmap has a blue streak along the main diagonal upper left to lower rightindicating that consecutive pairs of primes that have the same value mod 7 are strongly suppressed.
Away from the diagonal, in the upper right and lower left corners, we see a pastel checkerboard pattern.View and Download HP Color LaserJet Enterprise Mn troubleshooting manual online.
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