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Solving Families of Simultaneous Pell Equations

✍ Scribed by Michael A Bennett

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 185 KB
Volume: 67
Category: Article
ISSN: 0022-314X
DOI: 10.1006/jnth.1997.2188

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✦ Synopsis

If a and b are distinct positive integers then a previous result of the author implies that the simultaneous Diophantine equations

x 2 &az 2 =y 2 &bz 2 =1 possess at most 3 solutions in positive integers (x, y, z). On the other hand, there are infinite families of distinct integers (a, b) for which the above equations have at least 2 positive solutions. For each such family, we prove that there are precisely 2 solutions, with the possible exceptions of finitely many pairs (a, b). Since these families provide essentially the only pairs (a, b) for which the above equations are known to have more than a single solution (in positive (x, y, z)), this lends support to the conjecture that the number of such solutions to the above equations is 2 in all cases.

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