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On the size of the fundamental solution of the Pell equation

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Published/Copyright: May 15, 2014

Abstract

We give a lower bound for the cardinality of the set of Dx for which the fundamental solution of the Pell equation t2-Du2=1 is less than D12+α where α is any fixed constant such that α1. A second result, based on an hypothesis concerning short exponential sums, goes in the direction of a conjecture due to C. Hooley.

Funding statement: This research was partially supported by Institut Universitaire de France. The author thanks this institution.

The author thanks the referees very warmly. The exposition in this paper greatly benefited from their comments, questions and suggestions.

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Received: 2011-4-15
Revised: 2014-3-20
Published Online: 2014-5-15
Published in Print: 2016-8-1

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