Irreducibility of polynomials and arithmetic progressions with equal products of terms

In some fundamental papers Davenport, Lewis and Schinzel [DLS], Schinzel [Sch1, Sch3] and Fried [Fr1, Fr2, Fr3] have shown how irreducibility criteria for polynomials f(X) - g(Y) in combination with results of Runge or Siegel can be used to prove the finiteness of the solutions of the corresponding diophantine equation f(x) = g(y) in integers a, y. In the present paper we are particularly interested in the case f(X) = X(X + d(1)) ... (X + (m - 1)d(1)), g(Y) = Y(Y + d(2)) ... (Y + (n - 1)d(2)), i.e. the diophantine equation x(x + d(1)) ... (x + (m - 1)d(1)) = y(y + d(2)) ... (y + (n - 1)d(2)). (0.1) We first give some history on this equation and indicate how results for this equation can be derived from general irreducibility theory in the literature. Then we give direct proofs of the results using only basic facts on algebraic curves. 1991 Mathematics Subject Classification: 11D57.