Pi Mu Epsilon - Mathematical Association of America

Thursday MAA Session #6 August 2, 2012
MAA Session #6
Room: Meeting Room P
8:30A.M. – 10:25A.M.
8:30–8:45
Beal’s Conjecture, Lesser-known Brother of Fermat’s Last Theorem
Jakob Weisblat
Kent State University
Most people know a certain amount about Fermat’s Last Theorem (a n + b n = c n ; n ∈ Z >2 ) and its
history. However, there is another, more general, conjecture, proposed in the 20th century, that still
has not been proved or disproved. Beal’s Conjecture, also known as the Tijdeman-Zagier conjecture,
states that all solutions to the equation a x + b y = c z (x, y, z ∈ Z >2 ) have a, b, and c coprime.
This talk will discuss a personal search for patterns in the coprime solutions, some general coprime
solutions, and finally some possible proof angles and progress of research in the last 20 years.
8:50–9:05
Determining the Primality of a Given Integer is Easy
Tim Shaffer
Youngstown State University
It is well known that determining the prime factorization of a given integer can be quite computationally
expensive. In fact, outside of quantum computing, the most efficient factorization
algorithm known runs a little faster than in exponential time. What may not be so well-known is
that polynomial time algorithms exist that can determine the primality of a given integer. In this
presentation the Agrawal, Kayal, and Saxena (AKS) algorithm and the mathematics behind the
method are given, along with a comparison with other techniques, such as non-deterministic tests
for “probable primes”. Consideration for how this algorithm can be implemented and applied to
the search for prime numbers will also be discussed.
9:10–9:25
A Constructive Proof of the Cubic Case of Kronecker-Weber
Michael Mudarri
Hood College
The Kronecker-Weber theorem, first proved at the end of the 19 th century, states that any abelian
extension of the rational numbers Q is contained in a cyclotomic extension of Q. Let f be a cubic
polynomial with rational coefficients whose discriminant is a perfect square in Q. The Kronecker-
Weber theorem implies that the roots of f can be expressed as cyclotomic numbers, i.e. as Q-linear
combinations of roots of unity. The usual proofs of the theorem are not evidently constructive. I
will discuss an algorithm for constructing a representation of the roots of f as cyclotomic numbers
using the cubic formula and classical facts from the theory of cyclotomy.
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