Primes are distributed in a funny way:

They appear to have a random but non-random distribution. I don’t think anyone claims to understand it. We would expect prime density – number of primes per so many integers– to decrease as integers get larger, and that is the way it is. However,as we shall show, the curve is not uniform: there are clumps of primes.

Here are the primes in the integer group: 1 to 101:

##  [1]  2  3  5  7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83
## [24] 89 97
## There are  25  of them.

Here is scatter-point plot of them. Note that the green lines in this and subsequent plots are only there so you can follow the patterns more easily. The lines DO NOT indicate that this is a plot of a continuous function. Clearly it is discontinuous. The x_axis is integers or blocks of integers, and the y_axis is the number of primes associated with that block of integers. In all the plots there will be 100 blocks. In this first plot of integers 1 through 100, each block is only one integer, so the y_axis values will be 0 (that integer not a prime), or 1 (that integer is a prime).

## [1] "SUMMARY"
## Lowest Prime =  2 ;   Largest Prime =  97 ;   total number of primes:  25 ;  Mean Prime= 42.4
##    Integer Range= 1 to 101 :  Integer group size= 1

Here is a plot of prime density in integers ranging from 200 to 300:

## [1] "SUMMARY"
## Lowest Prime =  211 ;   Largest Prime =  293 ;   total number of primes:  16 ;  Mean Prime= 253
##    Integer Range= 200 to 300 :  Integer group size= 1

16 primes instead of 25. As one would expect, there are fewer primes in this group of 100 integers than in the previous group of 100 integers.

Let’s see what happens if we plt the density of primes in integer range 1 to 1000:

## [1] "SUMMARY"
## Lowest Prime =  2 ;   Largest Prime =  997 ;   total number of primes:  168 ;  Mean Prime= 453.1369
##    Integer Range= 1 to 1001 :  Integer group size= 10

The number of primes per group (10 integers per group) ranges from 0 to 4. (because there are 10 integers per group instead of 1). Can you see a pattern emerging, or is it your imagination?

The groups with no primes are not concentrated near the high end as we might expect. There is one group a little before the group beginning with the integer 800 that has four primes. It is immediately followed by a group that has one prime, and then no primes.

Let’s try plotting the numbers of primes in the integers ranging from 1000 to 10000:

## [1] "SUMMARY"
## Lowest Prime =  1009 ;   Largest Prime =  9973 ;   total number of primes:  1061 ;  Mean Prime= 5334.844
##    Integer Range= 1000 to 10000 :  Integer group size= 90

Interesting. In the 100 groups of 90 integers each, one group contains 16 primes. The least primes in a group is 7.

Lets see how larger integer primes are distributed:

## [1] "SUMMARY"
## Lowest Prime =  10007 ;   Largest Prime =  249989 ;   total number of primes:  20815 ;  Mean Prime= 125740.8
##    Integer Range= 10000 to 250000 :  Integer group size= 2400

We can see that the number of primes per group is trending downwards, but in an interesting irregulart way. Don’t you think?