Pattern formation Statistics on Fermat Quotients
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915354634092544 |
|---|---|
| author | Cobeli, Cristian Zaharescu, Alexandru Zhang, Zhuo |
| author_facet | Cobeli, Cristian Zaharescu, Alexandru Zhang, Zhuo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17684 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pattern formation Statistics on Fermat Quotients Cobeli, Cristian Zaharescu, Alexandru Zhang, Zhuo Number Theory Primary 11B99, Secondary 11A07 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. |
| title | Pattern formation Statistics on Fermat Quotients |
| topic | Number Theory Primary 11B99, Secondary 11A07 |
| url | https://arxiv.org/abs/2506.17684 |