# Polyadic Rings of p-Adic Integers

## Abstract

**:**

## 1. Introduction

## 2. $\left(\mathit{m},\mathit{n}\right)$-Rings of Integer Numbers from Residue Classes

**Example 1.**

## 3. Representations of p-Adic Integers

## 4. $\left(\mathit{m},\mathit{n}\right)$-Rings of p-Adic Integers

**Definition 1.**

**Definition 2.**

**Remark 1.**

**Lemma 1.**

**Proof.**

**Lemma 2.**

**Proof.**

**Theorem 1.**

**Proof.**

## 5. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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${\mathit{a}}_{\mathit{q}}$∖${\mathit{b}}_{\mathit{q}}$ | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|

1 | $\begin{array}{c}m=\mathit{3}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ | $\begin{array}{c}m=\mathit{4}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ | $\begin{array}{c}m=\mathit{5}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ | $\begin{array}{c}m=\mathit{6}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ | $\begin{array}{c}m=\mathit{7}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ | $\begin{array}{c}m=\mathit{8}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ | $\begin{array}{c}m=\mathit{9}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ | $\begin{array}{c}m=\mathit{10}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ | $\begin{array}{c}m=\mathit{11}\\ n=\mathit{2}\\ I=1\\ J=0\end{array}$ |

2 | $\begin{array}{c}m=\mathit{4}\\ n=\mathit{3}\\ I=2\\ J=2\end{array}$ | $\begin{array}{c}m=\mathit{6}\\ n=\mathit{5}\\ I=2\\ J=6\end{array}$ | $\begin{array}{c}m=\mathit{4}\\ n=\mathit{3}\\ I=1\\ J=1\end{array}$ | $\begin{array}{c}m=\mathit{8}\\ n=\mathit{4}\\ I=2\\ J=2\end{array}$ | $\begin{array}{c}m=\mathit{10}\\ n=\mathit{7}\\ I=2\\ J=14\end{array}$ | $\begin{array}{c}m=\mathit{6}\\ n=\mathit{5}\\ I=1\\ J=3\end{array}$ | |||

3 | $\begin{array}{c}m=\mathit{5}\\ n=\mathit{3}\\ I=3\\ J=6\end{array}$ | $\begin{array}{c}m=\mathit{6}\\ n=\mathit{5}\\ I=3\\ J=48\end{array}$ | $\begin{array}{c}m=\mathit{3}\\ n=\mathit{2}\\ I=1\\ J=1\end{array}$ | $\begin{array}{c}m=\mathit{8}\\ n=\mathit{7}\\ I=3\\ J=312\end{array}$ | $\begin{array}{c}m=\mathit{9}\\ n=\mathit{3}\\ I=3\\ J=3\end{array}$ | $\begin{array}{c}m=\mathit{11}\\ n=\mathit{5}\\ I=3\\ J=24\end{array}$ | |||

4 | $\begin{array}{c}m=\mathit{6}\\ n=\mathit{3}\\ I=4\\ J=12\end{array}$ | $\begin{array}{c}m=\mathit{4}\\ n=\mathit{2}\\ I=2\\ J=2\end{array}$ | $\begin{array}{c}m=\mathit{8}\\ n=\mathit{4}\\ I=4\\ J=36\end{array}$ | $\begin{array}{c}m=\mathit{10}\\ n=\mathit{4}\\ I=4\\ J=28\end{array}$ | $\begin{array}{c}m=\mathit{6}\\ n=\mathit{3}\\ I=2\\ J=6\end{array}$ | ||||

5 | $\begin{array}{c}m=\mathit{7}\\ n=\mathit{3}\\ I=5\\ J=20\end{array}$ | $\begin{array}{c}m=\mathit{8}\\ n=\mathit{7}\\ I=11\\ J=11,160\end{array}$ | $\begin{array}{c}m=\mathit{9}\\ n=\mathit{3}\\ I=5\\ J=15\end{array}$ | $\begin{array}{c}m=\mathit{10}\\ n=\mathit{7}\\ I=5\\ J=8680\end{array}$ | $\begin{array}{c}m=\mathit{3}\\ n=\mathit{2}\\ I=1\\ J=2\end{array}$ | ||||

6 | $\begin{array}{c}m=\mathit{8}\\ n=\mathit{3}\\ I=6\\ J=30\end{array}$ | $\begin{array}{c}m=\mathit{6}\\ n=\mathit{2}\\ I=3\\ J=3\end{array}$ | |||||||

7 | $\begin{array}{c}m=\mathit{9}\\ n=\mathit{3}\\ I=7\\ J=42\end{array}$ | $\begin{array}{c}m=\mathit{10}\\ n=\mathit{4}\\ I=7\\ J=266\end{array}$ | $\begin{array}{c}m=\mathit{11}\\ n=\mathit{5}\\ I=7\\ J=1680\end{array}$ | ||||||

8 | $\begin{array}{c}m=\mathit{10}\\ n=\mathit{3}\\ I=8\\ J=56\end{array}$ | $\begin{array}{c}m=\mathit{6}\\ n=\mathit{5}\\ I=4\\ J=3276\end{array}$ | |||||||

9 | $\begin{array}{c}m=\mathit{11}\\ n=\mathit{3}\\ I=9\\ J=72\end{array}$ |

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Duplij, S.
Polyadic Rings of *p*-Adic Integers. *Symmetry* **2022**, *14*, 2591.
https://doi.org/10.3390/sym14122591

**AMA Style**

Duplij S.
Polyadic Rings of *p*-Adic Integers. *Symmetry*. 2022; 14(12):2591.
https://doi.org/10.3390/sym14122591

**Chicago/Turabian Style**

Duplij, Steven.
2022. "Polyadic Rings of *p*-Adic Integers" *Symmetry* 14, no. 12: 2591.
https://doi.org/10.3390/sym14122591