  We show that "The product of consecutive integers is never a power" and
several results by Schinzel and Tijdeman on the solutions of the equation
$y^m=P(x)$, for $m>1$, $y>1$, and $P(x)$ a polynomial with rational
coefficients and with at least two distinct zeros, hold in a weak fragment of
Peano Arithmetic, $PA^-$, which lacks any kind of induction, and whose models
are the positive cones of discretely ordered rings.
