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 Real and Complex Techniques in Harmonic Analysis from the Point of View of Covariant Transform

This note reviews complex and real techniques in harmonic analysis. We describe a common source of both approaches rooted in the covariant transform generated by the affine group. Keywords: wavelet, coherent state, covariant transform, reconstruction formula, the affine group, ax+b-group, square integrable representations, admissible vectors, Hardy space, fiducial operator, approximation of the identity, maximal functions, atom, nucleus, atomic decomposition, Cauchy integral, Poisson integral, Hardy--Littlewood maximal functions, grand maximal function, vertical maximal functions, non-tangential maximal functions, intertwining operator, Cauchy-Riemann operator, Laplace operator, singular integral operator, SIO, boundary behaviour, Carleson measure.Comment: 31 pages, AMS-LaTeX, no figures; v2: a major revision, sections on representations of the ax+b group and transported norms are added; v3: major revision: an outline section on complex and real variables techniques are added, numerous smaller improvements; v4: minor correction