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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
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