On the Euler function of repdigits

Florian Luca

Czechoslovak Mathematical Journal (2008)

  • Volume: 58, Issue: 1, page 51-59
  • ISSN: 0011-4642

Abstract

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For a positive integer n we write φ ( n ) for the Euler function of n . In this note, we show that if b > 1 is a fixed positive integer, then the equation φ x b n - 1 b - 1 = y b m - 1 b - 1 , where x , y ∈ { 1 , ... , b - 1 } , has only finitely many positive integer solutions ( x , y , m , n ) .

How to cite

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Luca, Florian. "On the Euler function of repdigits." Czechoslovak Mathematical Journal 58.1 (2008): 51-59. <http://eudml.org/doc/31198>.

@article{Luca2008,
abstract = {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 $(x,y,m,n)$.},
author = {Luca, Florian},
journal = {Czechoslovak Mathematical Journal},
keywords = {Euler function; prime; divisor; Euler function; prime; divisor},
language = {eng},
number = {1},
pages = {51-59},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the Euler function of repdigits},
url = {http://eudml.org/doc/31198},
volume = {58},
year = {2008},
}

TY - JOUR
AU - Luca, Florian
TI - On the Euler function of repdigits
JO - Czechoslovak Mathematical Journal
PY - 2008
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 58
IS - 1
SP - 51
EP - 59
AB - 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 $(x,y,m,n)$.
LA - eng
KW - Euler function; prime; divisor; Euler function; prime; divisor
UR - http://eudml.org/doc/31198
ER -

References

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  1. Existence of primitive divisors of Lucas and Lehmer numbers (with an appendix by M. Mignotte), J. Reine Angew. Math. 539 (2001), 75–122. (2001) MR1863855
  2. On the numerical factors of the arithmetic forms α n ± β n , Ann. Math. 15 (1913), 30–70. (1913) MR1502458
  3. On the equation φ ( | x m + y m | ) = | x n + y n | , Indian J. Pure Appl. Math. 30 (1999), 183–197. (1999) MR1681596
  4. On the equation φ ( x m - y m ) = x n + y n , Irish Math. Soc. Bull. 40 (1998), 46–55. (1998) MR1635032
  5. Euler indicators of binary recurrent sequences, Collect. Math. 53 (2002), 133–156. (2002) MR1913514
  6. 10.2307/2975296, Amer. Math. Monthly 104 (1997), 871. (1997) DOI10.2307/2975296
  7. 10.1215/ijm/1256049659, Illinois J. Math. 20 (1976), 681–705 Zbl 0329.10035. (1976) Zbl0329.10035MR0419382DOI10.1215/ijm/1256049659
  8. On the distribution of amicable numbers, J. Reine Angew. Math. 293/294 (1977), 217–222. (1977) Zbl0349.10004MR0447087

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