Markoff numbers and ambiguous classes
نویسنده
چکیده
The Markoff conjecture states that given a positive integer c, there is at most one triple (a, b, c) of positive integers with a ≤ b ≤ c that satisfies the equation a2 + b2 + c2 = 3abc. The conjecture is known to be true when c is 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 discriminant d = 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 dividing d under which the conjecture holds. We also state a conjecture for the quadratic field Q( √ 9c2 − 4) that is equivalent to the Markoff conjecture for c.
منابع مشابه
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