| Title:
|
On the Euler function of repdigits (English) |
| Author:
|
Luca, Florian |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
58 |
| Issue:
|
1 |
| Year:
|
2008 |
| Pages:
|
51-59 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
For a positive integer $n$ we write $\phi (n)$ for the Euler function of $n$. In this note, we show that if $b>1$ is a fixed positive integer, then the equation \[ \phi \Big (x\frac{b^n-1}{b-1}\Big )=y\frac{b^m-1}{b-1},\qquad {\text{where}} \ x,~y\in \lbrace 1,\ldots ,b-1\rbrace , \] has only finitely many positive integer solutions $(x,y,m,n)$. (English) |
| Keyword:
|
Euler function |
| Keyword:
|
prime |
| Keyword:
|
divisor |
| MSC:
|
11A25 |
| idZBL:
|
Zbl 1174.11004 |
| idMR:
|
MR2402525 |
| . |
| Date available:
|
2009-09-24T11:53:29Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/128245 |
| . |
| Reference:
|
[1] Yu. Bilu, G. Hanrot and P. M. Voutier: Existence of primitive divisors of Lucas and Lehmer numbers (with an appendix by M. Mignotte).J. Reine Angew. Math. 539 (2001), 75–122. MR 1863855 |
| Reference:
|
[2] R. D. Carmichael: On the numerical factors of the arithmetic forms $\alpha ^n\pm \beta ^n$.Ann. Math. 15 (1913), 30–70. MR 1502458 |
| Reference:
|
[3] F. Luca: On the equation $\phi (|x^m+y^m|)=|x^n+y^n|$.Indian J. Pure Appl. Math. 30 (1999), 183–197. MR 1681596 |
| Reference:
|
[4] F. Luca: On the equation $\phi (x^m-y^m)=x^n+y^n$.Irish Math. Soc. Bull. 40 (1998), 46–55. MR 1635032 |
| Reference:
|
[5] F. Luca: Euler indicators of binary recurrent sequences.Collect. Math. 53 (2002), 133–156. MR 1913514 |
| Reference:
|
[6] F. Luca: Problem $10626$.Amer. Math. Monthly 104 (1997), 871. 10.2307/2975296 |
| Reference:
|
[7] K. K. Norton: On the number of restricted prime factors of an integer I.Illinois J. Math. 20 (1976), 681–705 Zbl 0329.10035. Zbl 0329.10035, MR 0419382, 10.1215/ijm/1256049659 |
| Reference:
|
[8] C. Pomerance: On the distribution of amicable numbers.J. Reine Angew. Math. 293/294 (1977), 217–222. Zbl 0349.10004, MR 0447087 |
| . |
