An *abc-triple* is a triple of pairwise coprime positive integers a, b, c with a + b = c. The *radical* r of such a triple is the product of the distinct prime numbers dividing abc, and the *quality* q is defined by q = (log c)/log r. For example, the triple given by a = 5, b = 27, c = 32 has r = 30 and q = (log 32)/log 30 = 1.018975235... . The *abc-conjecture* asserts that for any real number Q > 1, the number of abc-triples with quality greater than Q is finite. It is known that there do exist infinitely many abc-triples with quality greater than 1. The main subject of the lecture is an algorithm for listing, given a large integer N, all abc-triples with c at most N and quality greater than 1. As a byproduct, the algorithm yields an upper bound for the number of such triples, as a function of N.

One of M.C. Escher's most intriguing works depicts a man standing in a gallery who looks at a print of a city that contains the building that he is standing in himself. This picture, with the title Print Gallery, contains a mysterious white hole in the middle. In a paper of Hendrik Lenstra and the speaker in the April 2003 issue of the Notices of the AMS it is shown that well known mathematical results about elliptic curves imply that what Escher was trying to achieve in this work has a unique mathematical solution. This discovery opened up the way to filling the void in the print. With help from artists and computer scientists a completion of the picture was constructed at the Universiteit Leiden. The white hole turns out to contain the entire image on a smaller scale, which in the Dutch language is known as the Droste effect, after the Dutch chocolate maker Droste. In the talk the mathematics behind Escher's print and the process of filling the hole will be explained and visualized with computer animations. http://escherdroste.math.leidenuniv.nl/

Come to the Junior Number Theory organizational meeting.

Special Joint Meeting of Geometry and Number Theory