On blocks of arithmetic progressions with equal products
Let \(g\in \mathbb{Q} [X]\) be a monic polynomial of degree \(\mu\geq 2\) with simple real roots, and let \(f(X)= g^b(X)\) with \(b\in \mathbb{N}\). For given positive integers \(d_1, d_2, \ell,m\) with \(\ell <m\) and coprime, and \(\mu\leq m+1\) whenever \(m>2\), it is shown that the diophantine equation \[ f(x) f(x+d_1) \dots f\bigl(x+ (\ell k-1) d_1\bigr) =f(y) f(y+d_2) \dots f\bigl(y+ (\ell k-1) d_2 \bigr), \] with \(f(x+j d_1)\neq 0\) for \(0\leq j< \ell k\), has only finitely many solutions in integers \(x,y\) and \(k\geq 1\), except in the case \(m= \mu =2\), \(b= \ell= k=d_2 =1\), \(x= f(y)+y\). This effective (but not explicit) theorem extends a joint result of \textit{N. Saradha}, \textit{T. N. Shorey} and \textit{R. Tijdeman} [Acta Arith. 72, No. 1, 67-76 (1995; Zbl 0837.11015)].
arithmetic progressions with equal products
scientific article; zbMATH DE number 1132893