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Baker’s explicit \(abc\)-conjecture and applications. (English) Zbl 1276.11048

The precise statement of Baker’s explicit \(abc\)-conjecture is: Let \(a\), \(b\), \(c\) be pairwise coprime integers satisfying \(a+b=c\), then \[ c < {6 \over 5} N {(\log N)^\omega \over \omega !}, \] where \(N=N(abc)\) is the radical of the product \(abc\) and \(\omega\) is the number of distinct factors of \(abc\).
The authors show that this conjecture has very strong consequences for the classical Diophantine equations of Fermat-Catalan, Nagell-Ljunggren and Goormaghtigh. For example, the Nagell-Ljunggren equation \[ {x^n -1\over x-1}=y^q \] has the three well-known solutions \((x,y,n,q)=(3,11,5,2)\), \((7,20,4,2)\) and \(18,3,7,3)\). The authors prove that the above conjecture implies that these is no other solution.

MSC:

11D41 Higher degree equations; Fermat’s equation
11D45 Counting solutions of Diophantine equations
11D61 Exponential Diophantine equations
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