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Smooth solutions to the a b c equation: the x y z Conjecture

Jeffrey C. Lagarias, Kannan Soundararajan (2011)

Journal de Théorie des Nombres de Bordeaux

This paper studies integer solutions to the a b c equation A + B + C = 0 in which none of A , B , C have a large prime factor. We set H ( A , B , C ) = max ( | A | , | B | , | C | ) , and consider primitive solutions ( gcd ( A , B , C ) = 1 ) having no prime factor larger than ( log H ( A , B , C ) ) κ , for a given finite κ . We show that the a b c Conjecture implies that for any fixed κ < 1 the equation has only finitely many primitive solutions. We also discuss a conditional result, showing that the Generalized Riemann hypothesis (GRH) implies that for any fixed κ > 8 the a b c equation has infinitely many primitive solutions....

Solution to a problem of Bombieri

Andrew Granville (1993)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

We solve a problem of Bombieri, stated in connection with the «prime number theorem» for function fields.

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