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Start Over Creator Luca, Florian

1. Diophantine equations involving factorials

Creator:
Alzer, Horst and Luca, Florian
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Diophantine equation and factorial
Language:
English
Description:
We study the Diophantine equations (k!)n − k n = (n!)k − n k and (k!)n + k n = (n!)k + n k , where k and n are positive integers. We show that the first one holds if and only if k = n or (k, n) = (1, 2), (2, 1) and that the second one holds if and only if k = n.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

2. On the equation φ(|xm - ym |) = 2n

Creator:
Luca, Florian
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Euler function and Fermat primes
Language:
English
Description:
In this paper we investigate the solutions of the equation in the title, where φ 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.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public

3. On the Euler function of repdigits

Creator:
Luca, Florian
Format:
bez média and svazek
Type:
model:article and TEXT
Subject:
Euler function, prime, and divisor
Language:
English
Description:
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 $(x,y,m,n)$.
Rights:
http://creativecommons.org/publicdomain/mark/1.0/ and policy:public