Products of Consecutive Integers

In this paper, a number of results are deduced on the arithmeticstructure of products of integers in short intervals. By wayof an example, work of Saradha and Hanrot, and of Saradha andShorey, is completed by the provision of an answer to the questionof when the product of k out of k + 1 consecutive positive integerscan be an ‘almost’ perfect power. The main new ingredientin these proofs is what might be termed a practical method forresolving high-degree binomial Thue equations of the form axⁿ–byⁿ= ±1, based upon results from the theory of Galois representationsand modular forms. 2000 Mathematics Subject Classification 11D41,11D61.