On the last digits of tetrations of base $2^{k}$ and $5^{k}$

TL;DR

This paper proves formulas to determine the last digits of tetrations with bases 2^k and 5^k modulo 10^n, enabling faster computation of these rapidly growing expressions.

Contribution

It introduces new formulas for computing the last digits of tetrations with bases 2^k and 5^k modulo 10^n, applicable for different values of k.

Findings

Derived formulas for last digits of tetrations with bases 2^k and 5^k

Established stability of last digits beyond certain hyper-exponents

Facilitated faster computational methods for large tetrations

Abstract

In this paper will be proved the existence of a formula to reduce a tetration of base $2^{k}$ and $5^{k}$ $\mod 10^{n}$. Indeed, last digits of a tetration are the same starting from a certain hyper-exponent; In order to compute the last digits of those expressions we reduce them $\mod 10^{n}$. Lots of different formulas will be derived, for different cases of $k$ (where $k$ is the exponent of the base of the tetration). This kind of operation is fascinating, because the tetration grows very fast. But using these formulas we can actually have informations about the last digits of those expressions. It's possible to use these results on a software in order to reduce tetrations $\mod 10^{n}$ faster.