On values of a polynomial at arithmetic progressions with equal products. (English) Zbl 0837.11015

Let \(f \in \mathbb{Q} [X]\) be monic of degree \(\nu\). Further, let \(d_1, d_2, \ell,m\) with \(\ell < m\) and \(\text{gcd} (\ell, m)=1\) be given positive integers. The authors consider the equation \[ f(x) f(x+d_1) \dots f \bigl( x+(\ell k - 1) d_1 \bigr)=f(y) f(y+d_2) \dots f \bigl( y+(mk - 1) d_2 \bigr) \tag{1} \] in integers \(x,y\) and \(k \geq 2\) for which \(f(x+jd_1) \neq 0\) \((0 \leq j \leq \ell k - 1)\). In [Acta Arith. 68, 89-100 (1994; Zbl 0812.11023)] the authors considered the case \(f(X)=X\). The results obtained in this paper are extended in the present paper to the following:
Theorem. Under the conditions given above, equation (1) implies that \(k\) is bounded by an effectively computable (e.c.) number depending only on \(d_1, d_2, m\), and \(f\). Further, let \(f\) be a power of an irreducible polynomial. Then there exists an e.c. number \(C_3=C_3 (d_1, d_2, m,f)\) such that (1) implies that \[ \max \bigl\{ |x |, |y |, k \bigr\} \leq C_3 \]
unless \(\ell=1\), \(m=k=2\), \(d_1=2d^2_2\), \(f(X)=(X+r)^\nu\) with \(r \in \mathbb{Z}\), and \(x+r=(y+r) (y+r+3d_2)\).


11D61 Exponential Diophantine equations


Zbl 0812.11023
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