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Almost Squares and Factorisations in Consecutive Integers

Published online by Cambridge University Press:  04 December 2007

N. Saradha  and
T. N. Shorey
N. Saradha
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, India. e-mail: saradha@math.tifr.res.in
T. N. Shorey
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, India. e-mail: shorey@math.tifr.res.in
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Abstract

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We show that there is no square other than 122 and 7202 such that it can be written as a product of k−1 integers out of k(≥3) consecutive positive integers. We give an extension of a theorem of Sylvester that a product of k consecutive integers each greater than k is divisible by a prime exceeding k.

Keywords

consecutivediophantine equationselliptic equationsfactorisationprimes
Type
Research Article
Information
Compositio Mathematica , Volume 138 , Issue 1 , August 2003 , pp. 113 - 124
Copyright
© 2003 Kluwer Academic Publishers