## Measuring number symmetry and introduction of M(k)-tree

Starting from the last digit of every prime number (1,3,7,9), we then might iteratively append to numbers on the left side of the last digit (first added). This could be a way to create all (but divisible by 5) odd numbers. Now, to decrease the probability of constructing a non-prime number, we might think of a new to handle divisibility. We notice that divisibility by 3 and 9 is connected with the sums of digits. Due to the character of sum, we notice that for a number with digits abcdefg, when we cluster (ab)(cdef)(g) or (abc)(defg), it’s always the same about the remainders of the clusters, i.e. those cannot add up to a number divisible by 3 and 9. Not only the remainders but also potentially primiarility of the clusters also matters. Or, symmetry.  This post is just an indication that I would like to think about the symmetry of numbers. How to measure the symmetry of numbers to say that a number is prime or not. What could be the interesting clustering of the number?

As for the symmetry, I would like to introduce yet another abstract idea that came out of my dream. Take 1042341, lets paint its M(1)-tree.

1042341

Starting from the last digit as this is the digit that starts the number and might already tell us that we are not dealing with a prime. Lets start substraction: (n+1)th digit -nth digit.1. We have 1. (black)

The next digit is 4, so 4-1=3.(green, as positive)

2. So, we have 3.

Then, we have 3-4 = -1 (red, as negative)

3. So, we have -1. Then: -1,2,-4,1.

4. So, all in all, 1 black, 3 green, -1 red,-1 red, 2 green, -4 red, 1 green.

Painting it here:

That we could paint for all the numbers but for M(1) its length is the length of the number minus 1. In general case, we could cluster the number of digits into clusters of the same length and then add the digits forming the clusters. For instance, for 1042341, we have only one option as the number of digits is prime but for 10423411, we might have (10)(42)(34)(11) or (1042)(3411). In the first case the solution would be  23, 8,-32. In the second one: -2369.

And the question: does the primiarility of number of digits of a certain number influences the probability of a number being prime?