Senior Mathematics Seminar

Math 4513

Spring Semester 2007

The class test for this semester will take place on Friday March 30 this week. It will cover the material in chapters 2 (The Euclidean Algorithm) and 6 (The Gaussian Integers) of Stillwell's book. It will be an open-book style exam which means that you can refer to these chapters of Stillwell's book or class notes during the test however the use of calculators will not be permitted. Many of the problems on the test will be of a computational nature such as problems 2.2.1, 2.2.2, 2.3.1, 2.3.2, 2.5.3, 2.6.2, 2.6.3, 2.6.4, 6.1.1 and 6.2.4. You might also prepare for computational problems involving the Euclidean algorithm in Z[i] such as

• Find the greatest common divisor of 14+2i and 55-115i using the Euclidean algorithm.
• Use the result of the previous problem to determine a continued fraction expansion for (55-115i)/(14+2i).
• Find the greatest common divisor of 14+2i and 55-115i by first determining the prime factorizations of these two Gaussian integers.
• Use the Euclidean algorithm to show that 30+135i and -14+46i are relatively prime (that is, their gcd is 1) and find Gaussian integers z and w so that z(30+135i) + w(-14+46i)=1.

Here is a MATHEMATICA assignment due on Friday, 2/23: A twin prime is a prime p for which p+2 is also prime.

1. How many twin primes are there less than N where N= 100, 1000, 2000, 5000, and 10,000?
2. Create a ListPlot which graphically shows the percentage of primes that are twins up to 100000. (You can use a step size of 1000 for this.)
3. How many twin primes p between 1 and 10000 are there for which p+4 is also prime?
4. Define a MATHEMATICA function which computes the number of twin primes between two given integers M and N. (So the function takes M and N as input and returns the number of twin primes between M and N.)
5. Use the function in the previous part (or any other means) to find the first block of 1000 successive integers that contains no twin primes.

MATHEMATICA Problems from class on 2/12:

1. Replicate some of the experiments from MATHEMATICA demonstration 2/9 especially the commands that weren't discusses in class on Friday.
2. Copy and run the routines listed in determining prime gaps. Explain how each
3. Using ideas from (2):
   • determine the largest gap between primes less than 1000
   • find the first gap between primes with length 10, 20 and 100
   • create a list which gives the number of prime gaps equal to 1, 2, 4, 6, 8, 10 for the primes up to 1000.
4. Create title cell at the top of your notebook, including name and date. After cleaning up your notebook, print it out and turn it in.

On Monday, February 12 the class will meet (during its usual time) in the Computer Lab PHSC 230. We will work on using MATHEMATICA commands related to the Euler-Mascheroni constant and basic number theory involving primes and the prime counting function. To get started on this you might examine the Getting Started with Mathematica web page, and the MATHEMATICA demonstration from class on 2/9.

The course syllabus describes basic policies and goals for this course.

Some useful course links:

• The MacTutor History of Mathematics archive is the first place to look for reliable information concerning the history of mathematics. This site, which is maintained by the School of Mathematics and Statistics at the University of St. Andrews in Scotland, has a fantastic collection of biographies of famous mathematicians that includes useful bibliographies leading to other sources of information. The History Topics pages has a lot of interesting material also---the "Numbers and and Number Theory" section particularly has some interesting material for our class, including the "The Number e" page distributed in class.
• The Wikipedia biography of Leonhard Euler contains some interesting perspectives and links.
• Did you realize that April 15, 2007 is Euler's 300th birthday?

Math 4513, Spring 2007