Vol. 60, No. 1, 1975

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 294: 1
Vol. 293: 1  2
Vol. 292: 1  2
Vol. 291: 1  2
Vol. 290: 1  2
Vol. 289: 1  2
Vol. 288: 1  2
Vol. 287: 1  2
Online Archive
Volume:
Issue:
     
The Journal
Subscriptions
Editorial Board
Officers
Special Issues
Submission Guidelines
Submission Form
Contacts
Author Index
To Appear
 
ISSN: 0030-8730
The Diophantine equation Y (Y + m)(Y + 2m)(Y + 3m) = 2X(X + m)(X + 2m)(X + 3m)

S. Jeyaratnam

Vol. 60 (1975), No. 1, 183–187
Abstract

J. H. E. Cohn has shown that the equation of the tltle has only four pairs of nontrivial solutions in integers when m = 1. The object of this paper is to prove the following two theorems concerning the solutions of the equation of the title:

Theorem 1. The equation of the title has only four pairs of nontrivial solutions in integers given by X = 4m or 7m, Y = 5m or 8m when m is of the form

 r∏   si∏   tj
2    pi    qj

where r, si’s and tj’s are nonnegative integers, pi’s are positive primes 3,5 (mod 8) and qi’s are positive primes 1 (mod 8) such that

 (qj−1)∕4
2       ≡ − 1 (mod qi).

Theorem 2. The only positive integral solution of the equation of the title, for all positive integral values of m 30, is X = 4m, Y = 5m.

Mathematical Subject Classification
Primary: 10B10
Milestones
Received: 27 March 1974
Revised: 22 November 1974
Published: 1 September 1975
Authors
S. Jeyaratnam