Department of

# Mathematics

Seminar Calendar
for Mathematics Colloquium: Trjitzinsky Memorial Lectures events the year of Monday, November 5, 2018.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
     October 2018          November 2018          December 2018
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1  2  3  4  5  6                1  2  3                      1
7  8  9 10 11 12 13    4  5  6  7  8  9 10    2  3  4  5  6  7  8
14 15 16 17 18 19 20   11 12 13 14 15 16 17    9 10 11 12 13 14 15
21 22 23 24 25 26 27   18 19 20 21 22 23 24   16 17 18 19 20 21 22
28 29 30 31            25 26 27 28 29 30      23 24 25 26 27 28 29
30 31


Tuesday, November 27, 2018

4:00 pm in 314 Altgeld Hall,Tuesday, November 27, 2018

###### Carl Pomerance (Dartmouth College)

Abstract: How could there be something we don’t know about arithmetic? It would seem that subject was sewn up in third grade. But here’s a problem: What is the most efficient method for multiplication? No one knows. And here is another: How many different numbers appear in a large multiplication table? There are many more such problems, largely unsolved, and for which we could use some help!

A reception will be held from 5-6 pm in 239 Altgeld Hall following this first lecture.

Wednesday, November 28, 2018

4:00 pm in 245 Altgeld Hall,Wednesday, November 28, 2018

#### Random number theory

###### Carl Pomerance (Dartmouth College)

Abstract: No, this is not a talk about random numbers! Rather, we discuss the role of randomness in number theory, from Euler to the present. We'll visit the probabilistic background of Fermat's Last Theorem, the ABC conjecture, the Prime Number Theorem, the Riemann Hypothesis, Goldbach's Conjecture, and the Twin-Prime Conjecture, among other famous problems and results. Randomness has had a profound influence in computational number theory. In combinatorial number theory, the Probabilistic Method (best known in graph theory and combinatorics) is used to prove the existence of strange structures. In algebraic number theory the probability-based Cohen-Lenstra heuristics lead us to conjectures (and theorems too) about the distribution of algebraic number fields. We close with the famous Covering Congruences problem of Paul Erdös, which was recently settled with probabilistic tools.

Thursday, November 29, 2018

4:00 pm in 245 Altgeld Hall,Thursday, November 29, 2018

#### Primality testing: then and now

###### Carl Pomerance (Dartmouth College)

Abstract: The task is simply stated. Given a large integer, decide if it is prime or composite. Gauss wrote of this algorithmic problem (and the twin task of factoring composites) in 1801: "the dignity of science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated." Though progress with factoring composites has been steady and substantial, I think Gauss would be especially pleased with the enormous progress in primality testing, both in practice and in theory. In fact, one of the latest developments strangely and aptly employs a construct Gauss used to deal with ruler and compass constructions of regular polygons! This talk will present a survey of some of the principal ideas used in the prime recognition problem starting with the 19th century work of Lucas, to the 21st century work of Agrawal, Kayal, and Saxena, and beyond.