A systematic and complete proof of the existence and uniqueness of self-descriptive numbers

TL;DR

This paper provides a comprehensive and systematic proof of the existence and uniqueness of self-descriptive numbers across all bases, improving upon previous partial or trial-based methods.

Contribution

It offers the first complete, systematic proof of self-descriptive numbers' existence and uniqueness for all bases, including bases greater than 6.

Findings

Confirmed uniqueness of self-descriptive numbers for bases > 6

Developed a systematic proof scheme applicable to all bases

Eliminated trial-and-error methods in proving these properties

Abstract

All the already known results on self descriptive numbers, together with the demonstration of the uniqueness for bases greater than 6, are here obtained through a systematic scheme of proof and not trial and error. The proof is also complete for all the possible cases had been taken into account.