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Markoff numbers and ambiguous classes

Anitha Srinivasan (2009)

Journal de Théorie des Nombres de Bordeaux

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 a 2 + b 2 + c 2 = 3 a b c . 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 = 9 c 2 - 4 , every ambiguous form in the principal genus corresponds to a divisor of 3 c - 2 , then the conjecture is true. As a result, we obtain criteria in terms of...

Meilleures approximations diophantiennes simultanées et théorème de Lévy

Nicolas Chevallier (2005)

Annales de l’institut Fourier

D'après le théorème de Lévy, les dénominateurs du développement en fraction continue d'un réel croissent presque sûrement à une vitesse au plus exponentielle. Nous étendons cette estimation aux meilleures approximations diophantiennes simultanées de formes linéaires.

Mersenne numbers as a difference of two Lucas numbers

Murat Alan (2022)

Commentationes Mathematicae Universitatis Carolinae

Let ( L n ) n ≥ 0 be the Lucas sequence. We show that the Diophantine equation L n - L m = M k has only the nonnegative integer solutions ( n , m , k ) = ( 2 , 0 , 1 ) , ( 3 , 1 , 2 ) , ( 3 , 2 , 1 ) , ( 4 , 3 , 2 ) , ( 5 , 3 , 3 ) , ( 6 , 2 , 4 ) , ( 6 , 5 , 3 ) where M k = 2 k - 1 is the k th Mersenne number and n > m .

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