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Products of Consecutive Integers
Author(s) -
Bennett Michael A.
Publication year - 2004
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/s0024609304003480
Subject(s) - mathematics , mathematical proof , product (mathematics) , discrete mathematics , binomial (polynomial) , arithmetic , pure mathematics , statistics , geometry
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 = ±1, based upon results from the theory of Galois representations and modular forms. 2000 Mathematics Subject Classification 11D41, 11D61.