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An application named sft

Let P be the prime numbers set and let P(i) be the n-th prime, P(1) = 2. Let p be a prime consisting of n digits, we want to transform it in another n-digits prime. The process is similar to that used for circular primes and it will be very clear with an example: let p = 1997, now we shift ps digits one position left to obtain 997u, where u is an unknown digit.

We want 997u to be a prime so u must be in the set {1, 3, 7, 9} and in our example the only possibility is u = 3.

In general we wont obtain a unique result, in fact for example starting from 1187 well have 1871, 1873, 1877 and 1879 which are all primes.

On the other hand there are primes, such as 8713, for which none of the four possible numbers is prime.

Let p be a prime consisting of four digits abcd, lets consider the following set

{bcd1, bcd3, bcd7, bcd9};

then well erase the non prime elements from this set and well call N(p) the resulting set.

We call sft(p) the application from P to the elements of N(p).

In general, being p a n-digits prime, N(p) is given erasing the non prime elements from the set

{10 (p mod 10^(n 1)) + u}.

Construction of the adjacency matrix and evaluation of the entropy

The adjacency matrix C[n] of this application for primes up to n-digits is defined by the following rule: the (i, j) element is 1 if we can go from P(i) to P(j) by the application sft(P(i)), else the (i, j) element is 0.

Computing the entropy of this matrix gives us an idea of the complexity of the shift space resulting from the application sft.

Moreover entropy is an invariant (not complete) under conjugacy for this kind of spaces.

Using the Perron-Frobenius theory we can compute the entropy of the matrix C[n], called h(C[n]), defined by the following

h(C[n]) = log L,

where L is the largest eigenvalue of C[n].

This computation gives an unexpected result that is

h(C[2]) = h(C[3]) = h(C[4]) > 1.4066;

the corrisponding Perron eigenvalue is L > 2.5611.

This link will show you the sft application by a plot made starting from matrix C[4] in the following way: a point of coordinates (P(i), P(j)) is plotted corrisponding to the matrix elements equal to 1.
Moreover we can see the C[4] matrix plotting a point to represent an element equal to 1. This two plots are strangely similar.
A stochastic dynamical system

Iterating the application sft on the prime number set P we build a dynamical system on this set. To have a realization of such a system we must operate a choice everytime sft(p) contains more then an element.

If this choice is completely random we obtain a stochastic dynamical system as we can see from the following realizations: sometimes the orbits stop, for example
5279 2797
because sft(p) is empty; sometimes the orbits have no regularity and may continue indefinitely, for example
5279 2791 7919 9199 1997 9973 9739 7393 3931 9319 3191 1913 9137 ecc.

A deterministic dynamical system

If we state a way to choice inside the sft(p) set we'll see a completely different behaviour of the system: for example we can assume min(sft(p)) as iteration of p, now we have a deterministic dynamical system .

We'll have orbits that stop after a while as

2741 7411 4111 1117 1171

or more interesting orbits which become circular as

5279 2791 7919 9199 1993 9931 9311 3119 1193 1931 9311 3119 ...


1223 2237 2371 3719 7193 1931 9311 3119 1193 1931 9311 ...

In both the above cases the orbits become periodic as they reach the circular prime 9311. Moreover we can say that this two are the only possible behaviours for the points of such a dynamical system because it is a finite deterministic system.

The sft graphic contains every dynamical system that can be made by the C[4] matrix.

The author is Carla Chicchiero under direction of Prof. P.E. Ricci, University "La Sapienza", Rome.

For the theory about shift spaces see D. Lind and B. Marcus, "An Introduction to Symbolic Dynamics and Coding", Cambridge University Press.