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Markoff Numbers, Principal Ideals and Continued Fraction Expansions

✍ Scribed by J.O Button

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
152 KB
Volume
87
Category
Article
ISSN
0022-314X

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✦ Synopsis

Given any solution triple of natural numbers to the Markoff equation a 2 +b 2 + c 2 =3abc, an old problem asks whether the largest number determines the triple uniquely. We show this to be true in a range of cases by considering the factorisation of ideals in certain quadratic number fields, but also exhibit a counterexample for this approach when the question is widened to other numbers.

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Ankeny–Artin–Chowla Conjecture and Conti
Ankeny–Artin–Chowla Conjecture and Continued Fraction Expansion
✍ Ryūta Hashimoto 📂 Article 📅 2001 🏛 Elsevier Science 🌐 English ⚖ 115 KB

For any prime p congruent to 1 modulo 4, let (t+u -p)Â2 be the fundamental unit of Q(p). Then Ankeny, Artin, and Chowla conjectured that u is not divisible by p. In this paper, we investigate a certain relation between the conjecture and the continued fraction expansion of (1+p)Â2. Consequently, we