On blocks of arithmetic progressions with equal products

@article{Saradha1997,
abstract = {Let $f(X) \in \mathbb \{Q\}[X]$ be a monic polynomial which is a power of a polynomial $g(X ) \in \mathbb \{Q\}[X]$ of degree $\mu \ge 2$ and having simple real roots. For given positive integers $d_1, d_2, \ell , m$ with $\ell < m$ and gcd$(\ell ,m) = 1$ with $\mu \le m + 1$ whenever $m < 2$, we show that the equation\begin\{equation*\} f(x)f(x + d\_1) \cdots f(x + (\ell k - 1)d\_1) = f (y)f(y + d\_2) \cdots f(y + (mk - 1)d\_2) \end\{equation*\}with $f(x + jd_1) \ne 0$ for $0 \le j < \ell k$ has only finitely many solutions in integers $x, y$ and $k \ge 1$ except in the case\begin\{equation*\} m = \mu = 2, \ell = k = d\_2 = 1, f(X) = g(X),x = f(y) + y. \end\{equation*\}},
author = {Saradha, N.},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {arithmetic progressions with equal products; exponential diophantine equation},
language = {eng},
number = {1},
pages = {183-199},
publisher = {Université Bordeaux I},
title = {On blocks of arithmetic progressions with equal products},
url = {http://eudml.org/doc/248006},
volume = {9},
year = {1997},
}

TY - JOUR
AU - Saradha, N.
TI - On blocks of arithmetic progressions with equal products
JO - Journal de théorie des nombres de Bordeaux
PY - 1997
PB - Université Bordeaux I
VL - 9
IS - 1
SP - 183
EP - 199
AB - Let $f(X) \in \mathbb {Q}[X]$ be a monic polynomial which is a power of a polynomial $g(X ) \in \mathbb {Q}[X]$ of degree $\mu \ge 2$ and having simple real roots. For given positive integers $d_1, d_2, \ell , m$ with $\ell < m$ and gcd$(\ell ,m) = 1$ with $\mu \le m + 1$ whenever $m < 2$, we show that the equation\begin{equation*} f(x)f(x + d_1) \cdots f(x + (\ell k - 1)d_1) = f (y)f(y + d_2) \cdots f(y + (mk - 1)d_2) \end{equation*}with $f(x + jd_1) \ne 0$ for $0 \le j < \ell k$ has only finitely many solutions in integers $x, y$ and $k \ge 1$ except in the case\begin{equation*} m = \mu = 2, \ell = k = d_2 = 1, f(X) = g(X),x = f(y) + y. \end{equation*}
LA - eng
KW - arithmetic progressions with equal products; exponential diophantine equation
UR - http://eudml.org/doc/248006
ER -