| Title:
|
On the equation $\varphi (|x^m-y^m|)=2^n$ (English) |
| Author:
|
Luca, Florian |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
125 |
| Issue:
|
4 |
| Year:
|
2000 |
| Pages:
|
465-479 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper we investigate the solutions of the equation in the title, where $\phi$ is the Euler function. We first show that it suffices to find the solutions of the above equation when $m=4$ and $x$ and $y$ are coprime positive integers. For this last equation, we show that aside from a few small solutions, all the others are in a one-to-one correspondence with the Fermat primes. (English) |
| Keyword:
|
Euler function |
| Keyword:
|
Fermat primes |
| MSC:
|
11A25 |
| MSC:
|
11A51 |
| MSC:
|
11A63 |
| idZBL:
|
Zbl 0966.11002 |
| idMR:
|
MR1802295 |
| DOI:
|
10.21136/MB.2000.126267 |
| . |
| Date available:
|
2009-09-24T21:45:45Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126267 |
| . |
| Reference:
|
[1] R. D. Charmichael: On the numerical factors of arithmetic forms $\alpha^n ± \beta^n$.Ann. Math. 15 (1913-1914), 30-70. |
| Reference:
|
[2] F. Luca: Equations involving arithmetic functions of Fibonacci and Lucas numbers.Preprint. To appear in Fibo. Quart. Zbl 0941.11006, MR 1738646 |
| Reference:
|
[3] F. Luca: Pascal's triangle and constructible polygons.Preprint, Zbl 1006.11004 |
| . |
