On blocks of arithmetic progressions with equal products

N. Saradha

Journal de théorie des nombres de Bordeaux (1997)

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

Abstract

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Let f ( X ) ∈ ℚ [ X ] be a monic polynomial which is a power of a polynomial g ( X ) ∈ ℚ [ X ] of degree μ ≥ 2 and having simple real roots. For given positive integers d 1 , d 2 , ℓ , m with ℓ < m and gcd ( ℓ , m ) = 1 with μ ≤ m + 1 whenever m < 2 , we show that the equation f ( x ) f ( x + d 1 ) ⋯ f ( x + ( ℓ k - 1 ) d 1 ) = f ( y ) f ( y + d 2 ) ⋯ f ( y + ( m k - 1 ) d 2 ) with f ( x + j d 1 ) ≠ 0 for 0 ≤ j < ℓ k has only finitely many solutions in integers x , y and k ≥ 1 except in the case m = μ = 2 , ℓ = k = d 2 = 1 , f ( X ) = g ( X ) , x = f ( y ) + y .

How to cite

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Saradha, 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 $\ell &lt; m$ and gcd$(\ell ,m) = 1$ with $\mu \le m + 1$ whenever $m &lt; 2$, 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 $0 \le j &lt; \ell k$ 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 $\ell &lt; m$ and gcd$(\ell ,m) = 1$ with $\mu \le m + 1$ whenever $m &lt; 2$, 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 $0 \le j &lt; \ell k$ 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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  1. [1] A. Baker, Bounds for the solutions of the hyperelliptic equation, Proc. Camb. Phil. Soc.65 (1969), 439-444. Zbl0174.33803MR234912
  2. [2] A. Brauer and G. Ehrlich, On the irreducibility of certain polynomials, Bull.Amer. Math. Soc.52 (1946), 844-856. Zbl0060.04705MR17750
  3. [3] H.L. Dorwart and O. Ore, Criteria for the irreducibility of polynomials, Ann. of Math.34 (1993), 81-94. Zbl0006.00406MR1503098JFM59.0906.03
  4. [4] N. Saradha, T.N. Shorey and R. Tijdeman, On arithmetic progressions with equal products, Acta Arithmetica68 (1994), 89-100. Zbl0812.11023MR1302510
  5. [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. [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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