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On a conjecture of Mordell. (English) Zbl 1452.11030
The well-known Mordell’s conjecture states that if $$x+y\sqrt{p}$$ is the fundamental unit of $$\mathbb{Q}(\sqrt{p})$$ for a prime $$p$$ congruent to $$3$$ modulo $$4$$, then $$p$$ does not divide $$y$$. In other words, Mordell’s conjecture predicts that $$p$$ does not divide $$y$$ where $$(x,y)$$ is the fundamental solution to the equation $$x^2-py^2=1$$, when $$p \equiv 3 \pmod 4$$.
In the paper under review, the authors prove the following result. It is an equivalent criterion for non-divisibility of $$y$$ by $$p$$.
Theorem 1. Let $$x+y\sqrt{p}$$ denote the fundamental unit of the real quadratic field $$\mathbb{Q}(\sqrt{p})$$, where $$p$$ is a prime congruent to $$3$$ modulo $$4$$. Then $$p$$ divides $$y$$ if and only if $$p$$ divides $$h_{l/2-1}$$, where $$h_i$$ is the denominator of the $$i$$-th convergent of the continued fraction expansion of $$\sqrt{p}$$.
As a consequence, Theorem 1 allows the authors to confirm that Mordell’s conjecture holds when the regular continued fraction expansion of $$\sqrt{p}$$ has period length $$2, 4, 6,$$ or $$8$$. The proofs of their results purely rely on a clever combination of the properties of continued fractions and elementary techniques in number theory.

##### MSC:
 11D09 Quadratic and bilinear Diophantine equations 11A55 Continued fractions 11J70 Continued fractions and generalizations 11R11 Quadratic extensions 11R27 Units and factorization
##### Keywords:
continued fraction; period; fundamental unit
Full Text:
##### References:
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