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Keywords:
Euler function; prime; divisor
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 $(x,y,m,n)$.
References:
[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
[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
[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
[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
[5] F. Luca: Euler indicators of binary recurrent sequences. Collect. Math. 53 (2002), 133–156. MR 1913514
[6] F. Luca: Problem $10626$. Amer. Math. Monthly 104 (1997), 871. DOI 10.2307/2975296
[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. MR 0419382 | Zbl 0329.10035
[8] C. Pomerance: On the distribution of amicable numbers. J. Reine Angew. Math. 293/294 (1977), 217–222. MR 0447087 | Zbl 0349.10004
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