If you look closely, you will notice that the black lines in the image at the top right consist of individual dots.  These dots indicate when a number along the x-axis is divisible by a number along the y-axis.  The vertical line in the center corresponds to zero.  Clearly, zero is divisible by everything.  Thus, every row in this column contains a dot giving the impression of a solid black line.

The second image at the right is constructed in the same manner as the top image.  However, the vertical line in the middle corresponds to the number 327600.  The lines in this image are not solid because 327600 is not divisible by every number.  However, it is divisible by enough numbers to allow us to perceive the lines in the image.

The third image at the right is a magnified view of the middle image.  Notice how the vertical line in the middle as well as the diagonal lines radiate out from the point 327600.  This basic structure holds for every even number.  Notice that the diagonal lines to the left and right of the vertical line begin at 32759 and 327601 and that there is significant white space between these diagonals that the vertical line.  This white space illustrates graphically why we have twin primes.

In these images a prime number corresponds to columns that have two and only two dots.  There is a dot in the first row corresponding to the prime being divisible by 1.  There is also a dot in row p corresponding to the prime being divisible by itself.  Between the first row and row p, there are no dots signifying that the prime is not divisible by any other number.

Looking at the columns on either side of the vertical (highlighted in yellow) it is apparent that both columns have the potential of being prime.  In this particular example, there is the potential for column 327599 to contain dots only in rows 1 and 327599.  There is also the potential for column 327601 to contain dots only in rows 1 and 327601.  Should these conditions hold for both columns, then 327599 and 327601 would be twin primes (they are not twin primes.  327601 is divisible by 83).

If we assume that the dots in the white space are there by chance (not necessarily a correct assumption), then there would be some probability, P1, of a dot appearing in column 327599 in a row other than 1 or 327599.  Similarly, there would be a probability, P2, of a dot appearing in column 327601 in a row other than 1 or 327601.  In other words, there is a probability that 327599 is divisible by some number between 2 and 327598 and similarly for 327601.

On the flip side, there would be a probability that 327599 is not divisible by any number between 2 and 327598 and similarly for 327601.  What is this probability?  There is a 1/2 chance of 327599 not being divisible by 2.  There is a 2/3 chance of it not being divisible by 3.  There is a 3/4 chance of it not being divisible by 4, and so on.  We could calculate the probability of 327599 not being divisible by any of these numbers by calculating the product of all the associated probabilities.  However, the result would be in error due to the presence of conditional probabilities.

For example, if 327599 is not divisible by 3, then it surely is not divisible by 9.  Similarly, if 327599 is not divisible by 5, then it is not divisible by 25.  This dependency between divisors causes our probability calculation to be in error.

However, there are numbers that are independent of one another.  Not surprisingly, these are the prime numbers.  No prime number is divisible by another prime number.  If we know all the primes less than 327599, then we can use the probabilities associated with them to calculate the probability of 327599 not being divisible by them.  In other words, we can calculate the probability as 2/3 * 4/5 * 5/7 * 10/11 * ..., etc.  The square of this result gives us a small but non-zero probability that 327599 and 327601 are twin primes.

The fourth image on the right illustrates the use of these probabilities to estimate the number of primes less than p(n) where
p(n) is prime.  If we know all the primes less than p(n), then the probability can be calculated using the method just described.  This probability times p(n) gives a very good estimate of the number of primes less than p(n).  This is illustrated in the chart to the right where the estimated number of primes and actual number of primes are presented for the last 33 of the first 1000 primes.  The red line on the top represents the actual number of primes less than p(n) and the blue line on the bottom represents the number of primes as estimated using the above procedure.  The green line in the middle is calculated as p(n)/(log(p(n)) -1) from the Prime Number Theorem.  Notice how the shapes of the green and blue lines are very similar.

In summary, these images illustrate a structure within the divisors of the integers.  This structure of vertical and diagonal lines emanating from the even numbers illustrates the potential for an infinite number of twin primes.  However, it does not prove that this is, in fact, so.  This structure also suggests a way to calculate the probability of a number being prime that appears to be closely related to the method provided by the Prime Number Theorem.  This method can be stated precisely in the following equation where each p is a prime number:  where p(1) = 3.

 

 

 
 
Some thoughts about twin primes and the probability of a number being prime:

 

Visualizing the
Sieve of Eratosthenes
By David Cox

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