Strange behaviour of prime numbers

[fred_76]

Here is the graph showing the evolution of the proportions.

Graph 1 is the proportion of primes ending by 1, 3, 7 and 9 to the numbers of primes, by steps of 1 million primes.

   

The following graphs are showing the proportions of primes ending by 1, 3, 7 and 9 following primes ending by 1, 3, 7 and 9 to the number of primes ending by 1, 3, 7 and 9, again by steps of 1 million primes :

   

We would expect the curves to be in the range of 25% each, but it is not.

If we write [nm] to identify a couple of primes, the first ending by n followed by the second ending by m then :
- least frequent are the "double ending" 11, 33, 77, 99 with a % of occurence in the range of 17%.
- most frequent is 91 with an occurence of a bit less than 33%
- then 17, 13, 39 and 79 with about 30%
- and 19, 31, 37, 71, 73, 93, 97 between about 21% and 28%

The table for the 10 first million primes is as follows. It shows the proportion of primes ending by n, followed by a prime ending by m, to the number of primes ending by n :

Code:


nm    %
11    17.9%
13    30.2%
17    30.8%
19    21.1%
    
31    23.7%
33    16.9%
37    28.6%
39    30.8%
    
71    25.6%
73    27.3%
77    16.9%
79    30.3%
    
91    32.8%
93    25.6%
97    23.7%
99    17.8%