A Lower Bound in the abc Conjecture

Abstract

We show that there exists an infinite sequence of sums P: a+b=c of rational integers with large height compared to the radical:-2?Âe>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 K l . Academic PressKey Words: abc conjecture; good abc examples; the error term in the abc conjecture; sphere packing in n dimensions.

Lecture on the abc Conjecture and Some of Its Consequences

“…Further work on the abc conjecture for number fields (see [8]) are due to M. van Frankenhuijsen [73][74][75]77], N. Broberg [9], J. Browkin [11,12] …”

Two Restricted ABC Conjectures

“…because #{|ıa|, |ıb|, |ıc|} ≥ 2 always (with only one exception, see (11) below). The global radical is…”

“…The archimedean valuations are always counted (with weight 1 if ıK ⊂ R or 2 if ıK ⊂ R) because usually |ıa| or |ıb| = 1. The only exception is the sum of sixth roots of unity in K = Q[ √ −3 ], where we have defined the radical of the sum e πi/3 + e −πi/3 = 1 (11) as r K (e πi/3 : e −πi/3 : 1) = 2, but maybe it should be defined as 0 ?…”

Lower bounds for numbers of ABC-hits

“…Together with an idea of H.W. Lenstra, this C 0 = 4 √ 2 can even be replaced by C 0 < 4 · 2 0.2995 4 √ 2π/e; see [2]. Now define the counting function N(x) : R 0 → Z 0 by N(X) := ABC-hits (a, b, c) c X .…”

“…In [5] and [2] methods from the geometry of numbers together with versions of the prime number theorem with error term are used to arrive at the above mentioned results. In this paper these methods are applied to prove the following theorem.…”