On blocks of arithmetic progressions with equal products
Journal de théorie des nombres de Bordeaux, Volume 9 (1997) no. 1, pp. 183-199.

Let $f\left(X\right)\in ℚ\left[X\right]$ be a monic polynomial which is a power of a polynomial $g\left(X\right)\in ℚ\left[X\right]$ of degree $\mu \ge 2$ and having simple real roots. For given positive integers ${d}_{1},{d}_{2},\ell ,m$ with $\ell and gcd$\left(\ell ,m\right)=1$ with $\mu \le m+1$ whenever $m<2$, we show that the equation

 $f\left(x\right)f\left(x+{d}_{1}\right)\cdots f\left(x+\left(\ell k-1\right){d}_{1}\right)=f\left(y\right)f\left(y+{d}_{2}\right)\cdots f\left(y+\left(mk-1\right){d}_{2}\right)$
with $f\left(x+j{d}_{1}\right)\ne 0$ for $0\le j<\ell k$ has only finitely many solutions in integers $x,y$ and $k\ge 1$ except in the case
 $m=\mu =2,\ell =k={d}_{2}=1,f\left(X\right)=g\left(X\right),x=f\left(y\right)+y.$

Soit $f\left(X\right)\in ℚ\left[X\right]$ un polynôme qui est une puissance d’un polynôme $g\left(X\right)\in ℚ\left[X\right]$ de degré $\mu \ge 2$ et dont les racines réelles sont simples. Etant donnés les entiers positifs ${d}_{1},{d}_{2},\ell ,m$ satisfaisant $\ell pgcd$\left(\ell ,m\right)=1$ et $\mu \le m+1$ si $m<2$, nous démontrons que l’équation

 $f\left(x\right)f\left(x+{d}_{1}\right)\cdots f\left(x+\left(\ell k-1\right){d}_{1}\right)=f\left(y\right)f\left(y+{d}_{2}\right)\cdots f\left(y+\left(mk-1\right){d}_{2}\right)$
avec $f\left(x+j{d}_{1}\right)\ne 0$ pour $0\le j<\ell k$ ne possède qu’un nombre fini de solutions en les entiers $x,y$ et $k\ge 1$, excepté dans le cas
 $m=\mu =2,\ell =k={d}_{2}=1,f\left(X\right)=g\left(X\right),x=f\left(y\right)+y.$

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journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {183--199},
publisher = {Universit\'e Bordeaux I},
volume = {9},
number = {1},
year = {1997},
mrnumber = {1469667},
zbl = {0889.11010},
language = {en},
url = {http://www.numdam.org/item/JTNB_1997__9_1_183_0/}
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Saradha, N. On blocks of arithmetic progressions with equal products. Journal de théorie des nombres de Bordeaux, Volume 9 (1997) no. 1, pp. 183-199. http://www.numdam.org/item/JTNB_1997__9_1_183_0/

[1] A. Baker, Bounds for the solutions of the hyperelliptic equation, Proc. Camb. Phil. Soc. 65 (1969), 439-444. | MR | Zbl

[2] A. Brauer and G. Ehrlich, On the irreducibility of certain polynomials, Bull.Amer. Math. Soc. 52 (1946), 844-856. | MR | Zbl

[3] H.L. Dorwart and O. Ore, Criteria for the irreducibility of polynomials, Ann. of Math. 34 (1993), 81-94. | JFM | MR | Zbl

[4] N. Saradha, T.N. Shorey and R. Tijdeman, On arithmetic progressions with equal products, Acta Arithmetica 68 (1994), 89-100. | MR | Zbl

[5] N. Saradha, T.N. Shorey and R. Tijdeman, On values of a polynomial at arithmetic progressions with equal products, Acta Arithmetica 72 (1995), 67-76. | MR | Zbl

[6] T.N. Shorey, London Math. Soc. Lecture Note Series, Number Theory, Paris 1992-3, éd. Sinnou David, 215 (1995), 231-244. | MR | Zbl