Stochastic conjecture

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Stochastic conjecture is the believe that mathematics remains non-contradictory at the addition of the axiom below:

Let \(\mathbb N_+\) be set of natural numbers.

Let \(X \in \mathbb R\) be irrational number expressed with finite combination of rational numbers and elementary functions.

Let \(B \in \mathbb N_+\)

Let \(N_B\) be set of \(n\in \mathbb N_+\) such that \(0<n\le B\)

Let \(x_n \in \N_B \forall n\in \mathbb N_+\) such that

Let \(X\in \mathbb Z\)

Let \(x>X+\sum_{n=1}^M \frac{x_n}{B^n}\) for all \(M\in \mathbb N_+\)

Let \(x<X+\sum_{n=1}^M \frac{x_n}{B^n}+\frac{1}{B^M}\) for all \(M\in \mathbb N_+\)

Let \(F : (M\in \mathbb N_+, \vec{x} \in {N_B}^M) \mapsto \mathbb R\)

Then, for all \(M\in \mathbb N_+\), the following relation holds:

\(\displaystyle \lim_{ \begin{array}{c} R\in \mathbb N_+\\ R\rightarrow \infty \end{array} } \lim_{ \begin{array}{cc}K\in \mathbb N_+\\ K\rightarrow \infty \end{array}} \frac{1}{K} \sum_{k=0}^{K-1} F(M,\{x_{R+kM},x_{R+kM+1},x_{R+kM+2},..,x_{R+kM+(M-1)}\}) = \frac{1}{B^M} \sum_{\vec y \in {N_B}^M} F(M,\vec y)\)


The Stochastic Conjecture means, that in the positional representation of any physical irrational number, all the sequences of decimal digits are equally frequent.

Function \(F(M,\vec y)\) means some statistical procedure that tries to reveal any peculiarity in the sequence of \(M\) decimal digits stored in the array \(\vec y\).

The stochasticity means that any irrational number mentioned at the top, can be used as a source of pseudorandom numbers. The only way to reconstruct the sequence (or to show any regularities/peculiarities in it) is to guess a number representable as a linear function of \(X\) with rational coefficients. The set of such numbers has measure zero.

However, such an application menans that there is efficient way to evaluate \(X\) with arbitrary amount of decimal digits required.


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