98 KAM Mathematical Colloquium

# 98 KAM Mathematical Colloquium

## RANDOM NUMBER THEORY

Video and slides.
Tuesday November 8, 2016, 14:00, room S5 , second floor
KAM MFF UK
Malostranské nám. 25
118 00 Praha 1

## Abstract

No, it's not the theory of random numbers, though that's a part of it. Probabilistic methods have played a role in number theory for close to 90 years, if not longer. This lecture will highlight some of this history from Schoenberg to Turan, and the Erdos collaborations with Wintner and Kac. In the 1950's Erdos helped get the "probabilistic method" off the ground with his study of Sidon sequences. In the modern era, randomness has played an essential part in number-theoretic algorithms and in the way we think of the distribution of prime numbers. We will end our tour with the role of probability in the abc conjecture, the Erdos covering congruences problem, and the Catalan--Dickson conjecture.

## THE FIRST DYNAMICAL SYSTEM

Tuesday November 8, 2016, 10:30, room S5 , second floor
KAM MFF UK
Malostranské nám. 25
118 00 Praha 1

## Abstract

For a natural number n, let s(n) denote the sum of the positive divisors of n that are smaller than n. Introduced by Pythagoras 2500 years ago, it is perhaps the first function ever studied in mathematics. Additionally, Pythagoras suggested iterating s, finding some 1-cycles and 2-cycles. It has been conjectured (over a century ago) that there are no unbounded orbits, and there is a "counter" conjecture (from over 40 years ago), that most orbits starting at even numbers are unbounded. Amazingly, the first number in doubt is 276. I will report on some recent developments concerning the distribution of numbers in a cycle, numbers missing from the range of s, and some numerical and statistical results on the two conjectures.

## O přednášejícím

Profesor Carl Pomerance vystudoval na Harvardu, byl radnym profesorem na University of Georgia v Athens a pak na Dartmouth College. Byl hostem prednich mezinarodnich instituci (napr. IAS Princeton, MSRI Berkeley, Aisenstadt Chair, CRM). Je mezinarodne proslulym matematikem v teorii cisel --- zminme spolecny dukaz s Alfordem a Granvillem nekonecnosti poctu Carmichaelovych cisel - a publikoval temer v kazdem jejim odvetvi radu clanku (vcetne 17 praci s P. Erd\H osem). Za svou vedeckou cinnost ziskal radu oceneni (Chauvenet Prize MAA, Connat Prize 2001, AMS Fellow) a prednesl prednasku na ICM 1994 v Curychu. Profesor Pomerance je vynikajici prednasejici a je mimoradnou prilezitosti mit moznost vyslechnout v Praze hned dve jeho prednasky venovane ustredni oblasti jeho zajmu.