The Math Major Vol. 2, No. 4

The Math Major

CSU Fresno Mathematics Department

Vol 2. No. 4

Math Department Colloquium Schedule

The Math Department Colloquia are a series of talks intended for a general audience. Everyone is encouraged to attend and the talks are directed at people who have a reasonable comprehension of the topics in undergraduate mathematics. Come meet our undergraduates, graduate students and faculty as well as our distinguished guest speakers. The updated colloquia schedule can be found on the math departments web page. For more information contact Dr. Sean Cleary.

Date: Thursday, October 23th
Speaker: Dr. Glenn Appleby, Santa Clara University Department of Mathematics
Title: "Continued Fractions and Algebraic Numbers"
Time: 3:10-4pm
Location: San Ramon 6, Room 6
Abstract: In this talk I will describe a not-often-enough studied branch of mathematics called Continued Fraction representations of numbers. This is a different way of writing down real numbers that is similar to, but distinct from decimal expansions. We will use these representations to study what sort of numbers there are out there, and how we can tell them apart.

Math Alliance News

(From Dr. Velasquez) Students interested in a possible field trip to MSRI (Mathematical Sciences Research Institute) in Berkeley on October 22 should contact Dr. Velasquez.

Fermat's Last Theorem on TV

The Proof, a NOVA rebroadcast of the BBC program on the drama behind Andrew Wiles' proof of Fermat's Last Theorem, will be shown on PBS (local channel 18) on Tuesday evening, October 28 at 7 p.m. This is a rare opportunity to see modern mathematics explained to the average person. Invite your friends to watch. For a review of the original BBC broadcast, see the web page http://www.ams.org/publications/notices/199701/comm-granville.html.

Web Watch: Math Resources for Teachers Pre-K Through High School

The Math Forum (Swarthmore) has announced the addition of more than 50 new web pages for teachers. Formerly the K-12 Teachers' Place, this section of their site is now split by level, Pre-Kindergarten through High School. Check it out: http://forum.swarthmore.edu/teachers

A New Prime Record

On September 1, it was announced that Gordon Spence, using a program written by George Woltman, discovered that the number 2²⁹⁷⁶²²-1 (almost 900,000 digits long) is a prime number--the largest known prime to date. This number is one of a special class of prime numbers called Mersenne primes. Mersenne primes are prime numbers in the form 2^p - 1 where p itself is a prime number.

Gordon Spence is one of over 2000 volunteers world-wide participating in the Great Internet Mersenne Prime Search (GIMPS). This prime number is the second record prime found by the GIMPS project. Joel Armengaud discovered the previous largest known prime number last November. The GIMPS project was started by Woltman in early 1996.

The search for more Mersenne primes is already under way. There may be smaller, as yet undiscovered Mersenne primes, and there are certainly larger Mersenne primes waiting to be discovered. Anyone with a reasonably powerful personal computer can join GIMPS and become a big prime hunter. All the necessary software can be downloaded for free at

http://www.mersenne.org/prime.htm

Incidentally, it took Spence's 100 MHz Pentium computer 15 days to prove the number prime.

Another Math Major in the News

The winningest basketball coach in division 1 basketball history, Dean Smith of the University of North Carolina, (who recently announced his retirement) was a math major in college.

Problem Corner

Problem 2.3: Prove that the equation a³ + b³ = 2^c has no positive integer solutions with a>b.

Solution to Problem 2.3: If there are solutions, then clearly a and b have the same parity. If both are even, their cubes are divisible by 8; dividing through by 8 as many times as possible, we may assume a and b are both odd. Factor

a³+ b³ = (a+b)(a² - a b + b²) = 2^c,

which implies that each factor is a power of 2. Since a² - a b + b² is the sum of three odd numbers, it is odd; but the only power of 2 that is odd is 2⁰ = 1. Now a>b implies that a²- a b + b² = (a-b)² + a b > 1. Therefore there are no solutions.

Correct solutions were received from Anar Ahmedov and John Jamison.

New Problem

Problem 2.4: (Due no later than Thursday October 23, 4pm) The letters a, b and c represent different digits, a is a prime, and a-b = 4. If the number aaabbb is a prime, find the values of a, b and c.

Solutions may be delivered to the math department office (for Dr. Cusick) or by e-mail at larryc@csufresno.edu. There is a $75 dollar first prize and a $50 second prize to be awarded at the end of the semester to the student(s) who submit the most correct solutions.

CSU Fresno Math Department Home Page

California State University, Fresno