# On blocks of arithmetic progressions with equal products

Journal de théorie des nombres de Bordeaux (1997)

- Volume: 9, Issue: 1, page 183-199
- ISSN: 1246-7405

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topSaradha, N.. "On blocks of arithmetic progressions with equal products." Journal de théorie des nombres de Bordeaux 9.1 (1997): 183-199. <http://eudml.org/doc/248006>.

@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 -

## References

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- [4] N. Saradha, T.N. Shorey and R. Tijdeman, On arithmetic progressions with equal products, Acta Arithmetica68 (1994), 89-100. Zbl0812.11023MR1302510
- [5] N. Saradha, T.N. Shorey and R. Tijdeman, On values of a polynomial at arithmetic progressions with equal products, Acta Arithmetica72 (1995), 67-76. Zbl0837.11015MR1346806
- [6] T.N. Shorey, London Math. Soc. Lecture Note Series, Number Theory, Paris1992-3, éd. Sinnou David, 215 (1995), 231-244. Zbl0829.11015MR1345182

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