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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author = {N. Saradha},
title = {On blocks of arithmetic progressions with equal products},
journal = {Journal de Th\'eorie des Nombres de Bordeaux},
pages = {183--199},
publisher = {Universit\'e Bordeaux I},
volume = {9},
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year = {1997},
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zbl = {0889.11010},
mrnumber = {1469667},
language = {en},
url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.195/}
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N. Saradha. On blocks of arithmetic progressions with equal products. Journal de Théorie des Nombres de Bordeaux, Volume 9 (1997) no. 1, pp. 183-199. doi : 10.5802/jtnb.195. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.195/

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