# Stochastic conjecture

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:

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

## Meaning

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.

## References

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