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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....

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