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2012 | OriginalPaper | Buchkapitel

About the ABC Conjecture and an alternative

verfasst von : Machiel van Frankenhuijsen

Erschienen in: Number Theory, Analysis and Geometry

Verlag: Springer US

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Abstract

After a detailed discussion of the ABC Conjecture, we discuss three alternative conjectures proposed by Baker in 2004. The third alternative is particularly interesting, because there may be a way to prove it using the methods of linear forms in logarithms.

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Fußnoten
1
See [GdS07], which is also of interest to non-Dutch readers for a photographic reproduction of the relevant page.
 
2
In other words, (4) implies the classical result of Siegel and Mahler on the S-unit equation. The innovation of Stewart and Tijdeman was to use Baker’s theorem on linear forms in logarithms, generalized to p-adic logarithms, to make this result effective.
 
3
If ψ is not explicitly known, one would deduce that there could only be finitely many counterexamples to Fermat’s Last Theorem, but one would not know when to stop looking for one.
 
4
We have omitted from our table all abc sums with h > 50, since beyond a height of 50 our table is definitely not exhaustive and therefore useless. By November 2009, the project [LPS09] had resulted in an exhaustive search up to height 29. 9337 (i.e., up to c = 1013, apparently improved to 1020 [N09]). Schulmeiss has found some very large abc sums that satisfy (6), the largest of which has a height of 5, 114. Since these sums were not obtained by an exhaustive search, they are less useful to check different versions of the ABC Conjecture.
 
5
This criterion is closely related to the “merit”, see [GdS07,  dS09]. See also (7) below, which contains the same information as an inequality for the merit.
 
6
As alluded to in the introduction, the value 1 ∕ 2 may be related to the Riemann Hypothesis. Michel Waldschmidt pointed out to me that the most accessible approach to such a connection may be to construct a sequence of abc sums such that \(h(P) - r(P) \geq h{(P)}^{\theta -\varepsilon }\), given a hypothetical zero of the Riemann zeta function with real part θ > 1 ∕ 2.
 
7
If ω(c) is the least value among ω(a), ω(b) and ω(c), then \({\omega }_{\mathrm{max}} = \omega (a) + \omega (b)\) and \(\omega (abc) = {\omega }_{\mathrm{max}} + \omega (c) \leq{\omega }_{\mathrm{max}} + \frac{1} {2}(\omega (a) + \omega (b))\).
 
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Metadaten
Titel
About the ABC Conjecture and an alternative
verfasst von
Machiel van Frankenhuijsen
Copyright-Jahr
2012
Verlag
Springer US
DOI
https://doi.org/10.1007/978-1-4614-1260-1_9