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Smooth solutions to the \(abc\) equation: the \(xyz\) conjecture. (English. French summary) Zbl 1270.11032

Let \(A\), \(B\) and \(C\) be positive integers, and consider the Diophantine equation \(A+B=C\). One defines \[ H(A,B,C)=\max(A,B,C), \quad R(A,B,C)=\prod_{p \mid ABC} p, \quad S(A,B,C)=\max\{p; p \mid ABC\}, \] where \(p\) runs over prime numbers. A solution \((A,B,C)\) is called primitive when \(\gcd(A,B,C)=1\). The \(abc\) conjecture says that there exists a positive constant \(\kappa_1\) such that for any \(\varepsilon>0\) there are only finitely many primitive solutions to the equation \(A+B=C\) such that \[ R(A,B,C) \leq H(A,B,C)^{\kappa_1-\varepsilon}. \] To introduce the \(xyz\) conjecture let \((X,Y,Z)=(A,B,C)\), then the \(xyz\) conjecture says that there exists a positive constant \(\kappa_0\) such that the following hold.
a) For each \(\varepsilon>0\) there are only finitely many primitive solutions to the equation \(X+Y=Z\) such that \[ S(X,Y,Z) \leq (\log H(X,Y,Z))^{\kappa_0-\varepsilon}. \] b) For each \(\varepsilon>0\) there are infinitely many primitive solutions to the equation \(X+Y=Z\) such that \[ S(X,Y,Z) \leq (\log H(X,Y,Z))^{\kappa_0+\varepsilon}. \] The authors show that the \(abc\) conjecture implies that for any fixed \(\kappa<1\) there exist only finitely many primitive solutions to the equation \(X+Y=Z\) with \(S(X,Y,Z) \leq (\log H(X,Y,Z))^\kappa\). They also prove that the Generalized Riemann Hypothesis implies that for any \(\kappa>8\) the previous equation has infinitely many primitive solutions.

MSC:

11D45 Counting solutions of Diophantine equations
11N25 Distribution of integers with specified multiplicative constraints
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References:

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