In this paper, a number of results are deduced on the arithmetic structure of products of integers in short intervals. By way of an example, work of Saradha and Hanrot, and of Saradha and Shorey, is completed by the provision of an answer to the question of when the product of $k$ out of $k+1$ consecutive positive integers can be an ‘almost’ perfect power. The main new ingredient in these proofs is what might be termed a practical method for resolving high-degree binomial Thue equations of the form $ax^n - by^n = \pm 1$, based upon results from the theory of Galois representations and modular forms.