A Lower Bound in the abc Conjecture

We show that there exists an infinite sequence of sums P:a+b=c of rational integers with large height compared to the radical: h(P)⩾r(P)+4Klh(P)/logh(P) with Kl=2l/22π/e>1.517 for l=0.5990. This improves the result of Stewart and Tijdeman [9]. The value of l comes from an asymptotic bound for the...

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Veröffentlicht in:	Journal of number theory 2000-05, Vol.82 (1), p.91-95
1. Verfasser:	Van Frankenhuysen, Machiel
Format:	Artikel
Sprache:	eng
Schlagworte:	abc conjecture good abc examples sphere packing in n dimensions the error term in the abc conjecture
Online-Zugang:	Volltext
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Zusammenfassung:	We show that there exists an infinite sequence of sums P:a+b=c of rational integers with large height compared to the radical: h(P)⩾r(P)+4Klh(P)/logh(P) with Kl=2l/22π/e>1.517 for l=0.5990. This improves the result of Stewart and Tijdeman [9]. The value of l comes from an asymptotic bound for the packing density of spheres. We formulate our result such that improved knowledge of l immediately improves the value of Kl.
ISSN:	0022-314X 1096-1658
DOI:	10.1006/jnth.1999.2484