Markoff numbers and ambiguous classes

The Markoff conjecture states that given a positive integerc, there is at most one triple (a,b,c) of positive integers witha≤b≤cthat satisfies the equationa² +b² +c² = 3abc. The conjecture is known to be true whencis a prime power or two times a prime power. We present an elementary proof of this re...

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Bibliographic Details
Published inJournal de theorie des nombres de bordeaux Vol. 21; no. 3; pp. 757 - 770
Main Author SRINIVASAN, Anitha
Format Journal Article
LanguageEnglish
Published cedram 01.01.2009
Subjects
Discriminants
Equivalence relation
Integers
Mathematical theorems
Number theory
Numbers
Prime numbers
Uniqueness
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Summary:The Markoff conjecture states that given a positive integerc, there is at most one triple (a,b,c) of positive integers witha≤b≤cthat satisfies the equationa² +b² +c² = 3abc. The conjecture is known to be true whencis a prime power or two times a prime power. We present an elementary proof of this result. We also show that if in the class group of forms of discriminantd= 9c² − 4, every ambiguous form in the principal genus corresponds to a divisor of 3c−2, then the conjecture is true. As a result, we obtain criteria in terms of the Legendre symbols of primes dividingdunder which the conjecture holds. We also state a conjecture for the quadratic field ℚ ( 9 c 2 − 4 ) that is equivalent to the Markoff conjecture forc.
ISSN:1246-7405
2118-8572
DOI:10.5802/jtnb.701