Pietro Paparella

Notes on Number Theory and Discrete Mathematics

Print ISSN 1310–5132, Online ISSN 2367–8275

Volume 27, 2021, Number 3, Pages 119—122

DOI: 10.7546/nntdm.2021.27.3.119-122

**Download full paper: PDF, 146 Kb**

## Details

### Authors and affiliations

Pietro Paparella

*Division of Engineering and Mathematics, University of Washington Bothell
18115 Campus Way NE, Bothell, WA 98011, United States*

### Abstract

In this note, it is shown that if and are positive integers such that , then there is a *Perron number* such that . It is also shown that there is an aperiodic integer matrix such that .

### Keywords

- Perron number
- Fermat equation
- Integer matrix
- Aperiodic matrix

### 2020 Mathematics Subject Classification

- 11D41
- 15B36

### References

- Arnold, M., & Eydelzon, A. (2019). On matrix Pythagorean triples. The American Mathematical Monthly, 126(2), 158–160.
- Brenner, J. L., & de Pillis, J. (1972). Fermat’s equation
*A*+^{p}*B*=^{p}*C*for matrices of integers. Mathematics Magazine, 45, 12–15.^{p} - Brualdi, R. A., & Ryser, H. J. (1991). Combinatorial Matrix Theory. Cambridge: Cambridge University Press.
- Horn, R. A., & Johnson, C. R. (1990). Matrix Analysis. Cambridge: Cambridge University Press.
- Lind, D. A. (1983). Entropies and factorizations of topological Markov shifts. Bulletin of the American Mathematical Society, 9(2), 219–222.
- Lind, D. A. (1984). The entropies of topological Markov shifts and a related class of algebraic integers. Ergodic Theory and Dynamical Systems, 4, 283–300.

## Related papers

## Cite this paper

Paparella, P. (2021). Perron numbers that satisfy Fermat’s equation. Notes on Number Theory and Discrete Mathematics, 27(3), 119-122, doi: 10.7546/nntdm.2021.27.3.119-122.