Open Diophantine Problems

We collect a number of open questions concerning Diophantine equations, Diophantine Approximation and transcendental numbers. Revised version: corrected typos and added references.Comment: 58 pages. to appear in the Moscow Mathematical Journal vo. 4 N.1 (2004) dedicated to Pierre Cartie

LLL & ABC

This note is an observation that the LLL algorithm applied to prime powers can be used to find "good" examples for the ABC and Szpiro conjectures.Comment: 6 pages; record algebraic example included; final version, to appear in J. Number Theor

Smooth solutions to the abc equation: the xyz Conjecture

This paper studies integer solutions to the ABC equation A+B+C=0 in which none of A, B, C has a large prime factor. Set H(A,B, C)= max(|A|,|B|,|C|) and set the smoothness S(A, B, C) to be the largest prime factor of ABC. We consider primitive solutions (gcd(A, B, C)=1) having smoothness no larger than a fixed power p of log H. Assuming the abc Conjecture we show that there are finitely many solutions if p<1. We discuss a conditional result, showing that the Generalized Riemann Hypothesis (GRH) implies there are infinitely many primitive solutions when p>8. We sketch some details of the proof of the latter result.Comment: 21 pages, presented at 26th Journees Arithmetiques, 2009; v2 added new examples 1.2, updated references; v3 changed title, more examples added, notation changes, v4 corrects misprints in Conj. 3.1, Thm. 4.3 statement, 25 page

The strong ABCABC conjecture over function fields (after McQuillan and Yamanoi)

The $abc$ conjecture predicts a highly non trivial upper bound for the height of an algebraic point in terms of its discriminant and its intersection with a fixed divisor of the projective line counted without multiplicity. We describe the two independent proofs of the strong $abc$ conjecture over function fields given by McQuillan and Yamanoi. The first proof relies on tools from differential and algebraic geometry; the second relies on analytic and topological methods. They correspond respectively to the Nevanlinna and the Ahlfors approach to the Nevanlinna Second Main Theorem.Comment: 35 pages. This is the text of my Bourbaki talk in march 200