Pattern formation Statistics on Fermat Quotients

TL;DR

This paper investigates the seemingly irregular distribution of Fermat quotients modulo p, revealing that lines in the associated matrix behave randomly while overall spatial patterns exhibit regular statistical properties.

Contribution

It provides a novel analysis showing that Fermat quotient matrices combine random-like line behavior with regular spatial statistics, bridging number theory and statistical pattern analysis.

Findings

Lines in the Fermat quotient matrix behave like random sequences.

Spatial statistics of the matrix confirm natural regularities.

Contrasts between local randomness and global structure are demonstrated.

Abstract

Despite their simple definition as $\mathfrak{q}_p(b):=\frac{b^{p-1}-1}{p} \pmod p$, for $0\le b \le p^2-1$ and $\gcd(b,p)=1$, and their regular arrangement in a $p\times(p-1)$ Fermat quotient matrix $\mathtt{FQM}(p)$ of integers from $[0,p-1]$, Fermat quotients modulo $p$ are well known for their overall lack of regularity. Here, we discuss this contrasting effect by proving that, on the one hand, any line of the matrix behaves like an analogue of a randomly distributed sequence of numbers, and on the other hand, the spatial statistics of distances on regular $N$-patterns confirm the natural expectations.