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2012 | OriginalPaper | Chapter

About the ABC Conjecture and an alternative

Author : Machiel van Frankenhuijsen

Published in: Number Theory, Analysis and Geometry

Publisher: 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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previous chapter Harmonic representatives for cuspidal cohomology classes

next chapter Unifying themes suggested by Belyi’s Theorem

See [GdS07], which is also of interest to non-Dutch readers for a photographic reproduction of the relevant page.

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.

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.

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.

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.

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.

If ω(c) is the least value among ω(a), ω(b) and ω(c), then \({\omega }_{\mathrm{max}} = \omega (a) + \omega (b)\) and \(\omega (abc) = {\omega }_{\mathrm{max}} + \omega (c) \leq{\omega }_{\mathrm{max}} + \frac{1} {2}(\omega (a) + \omega (b))\).

B96.

A. Baker, Logarithmic forms and the abc-conjecture, in: Number theory (Diophantine, computational and algebraic aspects), Proceedings of the international conference, Eger, Hungary, July 29–August 2, 1996 (Györy, Kalman et al., ed.), de Gruyter, Berlin, 1998, 37–44.

B04.

A. Baker, Experiments on the abc-conjecture, Publ. Math. Debrecen 65/3-4 (2004), 253–260.

C73.

J. R. Chen, On the Representation of a Large Even Integer as the Sum of a Prime and the Product of at Most Two Primes, Sci. Sinica 16 (1973), 157–176.

C78.

J. R. Chen, On the Representation of a Large Even Integer as the Sum of a Prime and the Product of at Most Two Primes, II, Sci. Sinica 21 (1978), 421–430.

dS09.

B. de Smit, http://www.math.leidenuniv.nl/\~desmit/abc/, 2009.

GdS07.

G. Geuze, B. de Smit, Reken mee met ABC, Nieuw Archief voor de Wiskunde 5/8, no. 1, March 2007.

HW60.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, Oxford, 1960.

La05.

S. Lang, Questions about the error term of Diophantine inequalities, preprint, 2005.

LC90.

S. Lang and W. Cherry, Topics in Nevanlinna Theory, Lect. Notes in Math. 1433, Springer-Verlag, New York, 1990.

LPS09.

H. W. Lenstra, W. J. Palenstijn and B. de Smit, Reken Mee met ABC, http://www.rekenmeemetabc.nl/, 2009, and http://abcathome.com/.

Ma84.

R. C. Mason, Diophantine Equations over Functions Fields, London Math. Soc. LNS 96, Cambridge, 1984.

Mas02.

D. Masser, On abc and discriminants, Proc. Amer. Math. Soc. 130 (2002), 3141–3150.

N09.

A. Nitaj, http://www.math.unicaen.fr/\~nitaj/abc.html, 2009.

O88.

J. Oesterlé, Nouvelles approches du “Théorème” de Fermat, Sém. Bourbaki 1987–1988 no. 694, Astérisque 161–162 (1988), 165–186.

Si84.

J. H. Silverman, The S-unit equation over function fields, Math. Proc. Cambridge Philos. Soc. 95 (1984), no. 1, 3–4.

Sm93.

A. L. Smirnov, Hurwitz inequalities for number fields, St. Petersburg Math. J. 4 (1993), 357–375.

StTe.

C. L. Stewart, G. Tenenbaum, A refinement of the ABC Conjecture, preprint.

StTi86.

C. L. Stewart and R. Tijdeman, On the Oesterlé-Masser conjecture, Mh. Math. 102 (1986), 251–257.

StY01.

C. L. Stewart and K. Yu, On the abc conjecture, II, Duke Math. J. 108 (2001), 169–181.

Sto81.

W. W. Stothers, Polynomial identities and hauptmoduln, Quart. J. Math. Oxford (2) 32 (1981), 349–370.

vF95.

M. van Frankenhuijsen, Hyperbolic Spaces and the ABC Conjecture, thesis, Katholieke Universiteit Nijmegen, 1995.

vF00.

M. van Frankenhuijsen, A lower bound in the ABC Conjecture, J. of Number Theory 82 (2000), 91–95.

vF04.

M. van Frankenhuijsen, The ABC conjecture implies Vojta’s Height Inequality for Curves, J. Number Theory 95 (2002), 289–302.

vF06.

M. van Frankenhuijsen, ABC implies the radicalized Vojta height inequality for curves, J. Number Theory 127 (2007), 292–300.

Title: About the ABC Conjecture and an alternative
Author: Machiel van Frankenhuijsen
Publisher: Springer US
Book: Number Theory, Analysis and Geometry
Print ISBN: 978-1-4614-1259-5

Electronic ISBN: 978-1-4614-1260-1

Copyright Year: 2012
DOI: https://doi.org/10.1007/978-1-4614-1260-1_9

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