MPSolve

Trying to make a proof of the multiplicative persistence conjecture using a formal language.
The idea is to start by the proof of base 2 and then try to generalize it to other bases. We can either extend the proof in base 3 and prove that the multiplicative persistennce is 1 more than in base 2 then extends to all other bases. Or we can prove that the multiplicative persistence in base 3 doesn't have a majoration and then generalize it to all other bases. I guess that we can use the Solane conjecture for that but I don't know how to use it yet.

Neil Sloane mentions a more general conjecture: for any number base b, there exists a constant M(b) such that no integer expressed in this base b has a multiplicative persistence greater than M(b)1.

In our case we define P = 0 as the set of all digits and trying to define all other numbers, the multiplicative persistance is the first P that we can't define using P-1.

Base 2

Define base 2 as a language

S -> Z | O | O S
Z -> "0"
O -> "1"

Define the multiplicative operation (sca : multiplicative anihihilator, sci : multiplicative identity)

sca -> O | O S
sci -> Z | O S | T S

Define the different set of multiplicative persistence (P)

Cases for P -> P-1:
P = 0;
Z -> Z
O -> O

P = 1;
Osca -> Z
Osci-> O

P = 2;
Can't create a valid expression with P = 2, all numbers are described by P = 0 or P = 1

Textual proof

In base 2, a number is either P0, which means it is a digit, or it is P1, which means it is either Osca or Osci (Since a number cannot start with a zero), which results in P0.
Osca and Osci define all numbers greater than P0.
Therefore, we have defined all possible numbers in base 2 using P0 and P1.

Base 3

Work in progress, I don't know if the proof is valid yet, still don't understand it fully. https://g.co/bard/share/8266b324a4da

Other bases

TODO : Make a construction method for base 2 and then generalize it to other bases or prove that it is not possible.

Notes

Don't mind doing a pull request to improve this proof

References

French Wikipedia article on multiplicative persistence

English Wikipedia article on multiplicative persistence

Neil Sloane's article on multiplicative persistence

OEIS article on multiplicative persistence

The Multiplicative Persistence Conjecture Is True for Odd Targets (2021 research paper)

My older work with friends on this conjecture (french highschool math club article)