On arithmetic progression with equal products. (English) Zbl 0812.11023

Saradha, N.; Shorey, T. N.; Tijdeman, R.

Acta Arith. 68, No. 1, 89-100 (1994).

Let \(d_ 1\) and \(d_ 2\) be given integers. It is shown by an effective method that the equation \[ x(x+ d_ 1)\dots (x+ (L-1) d_ 1)= y(y+ d_ 2)\dots (y+ (M-1) d_ 2) \] in positive integers \(L>1\), \(M>1\), \(x\), \(y\) subject to \(L\neq M\) and \((L,M)\neq (2,4), (4,2)\) admits only finitely many solutions if (i) \(L\in \{2,4\}\) and \(M>2\) is given, or (ii) \(\text{gcd} (L,M)\) and \(L/M\) has a given ratio.
The cases \(L=M\) and \((L,M)= (2,4)\) or (4,2) have been treated in other papers and both admit infinitely many solutions.

Reviewer: R.Tijdeman (Leiden)

MSC:

11D61

Exponential Diophantine equations

Keywords:

exponential diophantine equation; effective method