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Invited Paper Session Abstracts - Numbers, Geometries, and Games: A Centenarian of Mathematics

Saturday, August 6, 1:00 p.m. - 3:10 p.m., Fairfield

Born nearly at the same time as the MAA, Richard Guy has had a tremendous impact on mathematics through his (continuing) work in number theory, geometry, and game theory. This session brings together friends an colleagues to talk about these mathematical areas, to celebrate Richard's achievements, and to mark his transition to his second century.

Organizers:
Steve Butler, Iowa State University
Barbara Faires, Westminster College

Sums of Unit Fractions

1:00 p.m. - 1:20 p.m.
Ron Graham, University of California San Diego

In this talk I will describe a number of results and open problems dealing with so-called Egyptian fractions, i.e., representations of rationals as sums of unit fractions with distinct denominators. These occur in some of the earliest known mathematical manuscripts (\(approx\)1650 B.C) and were a favorite topic of the late Paul Erdős. These are also covered in Richard Guy's Unsolved Problems in Number Theory and, in particular, it was the references there which were responsible for the most recent paper of Erdős (published within the last year).

Products of Farey Fractions

1:30 p.m. - 1:50 p.m.
Jeffrey Lagarias, University of Michigan

The Farey fractions of order N are the rational fractions in the unit interval which in lowest terms have denominators at most \(N\). Farey fractions have generally been studied additively, as in Problem F11 in Richard Guy's book, Unsolved Problems in Number Theory. We describe results arising from an REU project that studies them multiplicatively: How do products of all (nonzero) Farey fractions of order \(N\) behave, as a function of \(N\)? This work was done with Harsh Mehta (now a grad student at U. South Carolina).

Some Tiling Problems

2:00 p.m. - 2:20 p.m.
Steve Butler, Iowa State University

Many interesting mathematical problems arise from and can be related to problems involving tiling. We discuss a tiling problem with connections to well known integer sequences and also consider a result of tiling which blends geometry and number theory.

Fibonacci Plays Billiards, Again

2:30 p.m. - 2:50 p.m.
Elwyn Berlekamp, University of California Berkeley

One version of the classic traveling salesman problem seeks to determine whether or not, in any given graph, there exists a "Hamiltonian path" which traverses every node exactly once. In the general case, this problem is well-known to be NP Hard. In one interesting subclass of this problem, the nodes are taken to be the first \(N\) integers, \(\{1, 2, 3,\ldots,N\}\), where there is a branch between \(J\) and \(K\) if \(J+K\) is in a specified set \(S = \{S[1], S[2], S[3],\ldots,S[M]\}\). Or, given \(S\), for what values of \(N\) does a Hamiltonian path exist? How fast can the elements of \(S\) grow such that there exist solutions for infinitely many \(N\)?

The answer to the second question turns out to be a close relative of the Fibonacci numbers, for which we construct solutions by observing the path of a billiard ball which travels at \(45\) degree angles to the sides of its table. Using the same billiard ball methodology, we also find some particular solutions when \(S\) is the set of squares or the set of cubes.

Remarks

3:00 p.m. - 3:10 p.m.
Richard Guy, University of Calgary

Year: 
2016