Author: Eiko

Time: 2025-03-06 12:18:34 - 2025-03-06 12:18:34 (UTC)

Technical Bound For Log Inequality

The type of inequality we will encounter in this adventure is of the form

$$n\le A\lfloor {\log }_p^{+}n\rfloor +B$$

where $A,B\ge 0$ are constants, $p>1$ is a prime number, and ${\log }_p^{+}n=max(0,{\log }_{p}n)$ is the positive part of the base-$p$ logarithm of $n$. The goal is to obtain a bound on $n$ in terms of $A,B$ and $p$.