Observations regarding the repetition of the last digits of a tetration of generic base

TL;DR

This paper explores the repeating patterns of last digits in tetrations of generic bases, proposing a conjecture relating these patterns to the base's residue modulo 10 and its exponents, supported by tables and examples.

Contribution

It introduces a conjecture linking last digit repetition in tetrations to the base's residue and exponent structure, approaching a proof for a formula determining when last digits stabilize.

Findings

Repetition of last digits depends on base residue mod 10.

Proposed a formula for the minimal hyper-exponent where last digits stabilize.

Supported conjecture with tables and examples.

Abstract

This paper investigates the behavior of the last digits of a tetration of generic base. In fact, last digits of a tetration are the same starting from a certain hyper-exponent and in order to compute them we reduce those expressions $\mod 10^{n}$. Very surprisingly (although unproved) I think that the repetition of the last digits depend on the residue $\mod 10$ of the base and on the exponents of a particular way to express that base. Then I'll discuss about the results and I'll show different tables and examples in order to support my conjecture. We are very near to a proof for a formula which finds the minimum hyper-exponent $u$ of a tetration with a generic base $q$ such that the last $n$ digits of the tetration starting from the $u$-th one are the same.