The Diophantine equation u(u + 1)(u + 2)(u + 3) = v(v + 1)(v + 2)

David W. Boyd and Hershy Kisilevsky

Vol. 40 (1972), No. 1, 23–32

DOI: 10.2140/pjm.1972.40.23

Abstract

In this paper we demonstrate that the equation of the title has exactly three solutions in positive integers, namely: 1 ⋅ 2 ⋅ 3 ⋅ 4 = 2 ⋅ 3 ⋅ 4,2 ⋅ 3 ⋅ 4 ⋅ 5 = 4 ⋅ 5 ⋅ 6 and 19 ⋅ 20 ⋅ 21 ⋅ 22 = 55 ⋅ 56 ⋅ 57. The method of proof is to reduce the equation to the form y² = x³ − x + 1.

Mathematical Subject Classification

Primary: 10B10

Milestones

Received: 11 May 1970

Published: 1 January 1972