MAT 331: Computer-Assist Math Prob Solv
Spring 2010
schedule

## Schedule

Here is the evaluation form for the presentations (It has a time time stamp and you are only allowed to fill it in class).

The Wiki should be written in Blackboard. As usual Maple files should also be submitted in Blackboard. The description of Project 2 is here

Some interesting papers about the topics of the presentations can be found here.

An excelent lecture about primes: Terence Tao: Structure and Randomness in the Prime Numbers, UCLA

 Week HW due Tuesday Thursday May 5th Read topics for this week. Primality tests Twin primes; twin primes conjecture Mersenne primes, Fermat primes, Pepin's test. 8 April 26th Read topics for this week. Divisibility rules Fermat Little Theorem Pseudo - primes; Carmichel numbers Euclid's algorithm April 19th Read topics for this week. Work on Project 3 Wiki Add to the wiki one of the topics listed in project 2. Goldbach conjecture Infinitud of primes ; Density of primes Fundamental theorem of Arithmetic; amicable numbers.  Fermat Last Theorem; Sophie Germain primes April 12th Work on Project 3 Wiki Add to the wiki one of the topics listed in project 2. Cryptography Work on project 3. Cryptography Work on project 3. April 5th Project 2 Cryptography Work on project 3. Cryptography Work on project 3.

## Maple

1

Desirae, Danielle R

Lauren D.

Divisivility  rules

Divisibility rules in base 2,12 and 60.

Divisibility

Program to test divisibility using the algorithm

2

Jing-Gu

Rondell

Daisuke

Fermat little theorem.

Proof of Fermat Little Theorem

Program to compute the multiplicative order of elements of Z/nZ

3

Lauren D, Nicole M, Carmelina

Twin primes

Twin prime conjectures. Brun's theorem.

Proof that the sum of the reciprocal of the primes diverges.

Program to find twin primes.

Compare actual results with guessed or conjectured density

4

Michael C.

Sabil

Deema

Pseudo primes

Carmichel numbers

Prove that there are infinitely many pseudo primes with base a.

Programs to test pseudo primes.

Compare number of primes and number of pseudo-primes

5

Carson

Monica

Jose

There are infinitely many primes, density, probability that a given number is prime.

Filip Saidak's

Euclid proof

Other proofs of this result.

Comparison between  estimated and actual density of primes.

6

Anand

Sean

The Goldbach conjecture,

Patterns in Goldbach curve.

count goldbach pairs, do you see any patterns?

7

Ren,

Cristi,

Alaa,

Xixin

Primality tests (Strong Probable  Primes)

Determine probability that a number satisfies the prime test.

Write a program for primality test

8

Ceandra,

Misra,

Alexander M.

Mersenne primes

Fermat primes

Pepin's Test

Let p and q be odd primes. If p divides Mq, then p = 1 (mod q) and p = +/-1 (mod 8).

Generate mersene and fermat primes.

Pepin's test

9

Nicole C.

Ashley

Fermat last theorem

Sophie Germain prime.

Proof that  Let p = 3 (mod 4) be prime. 2p+1 is also prime if and only if 2p+1 divides Mp.

find Sophie Germain primes

Examples of the theorem.

10

Ken

Chantilly

Alexander N

The fundamental theorem of arithmetic

Amicable numbers

Prove of the result. Examples in other fields.

find amicable numbers

11

Davanjit, Konstantin

Euclid algorithm

The probability that two given numbers are relatively prime.

experiment with the result you proved.

## First part of the course schedule

 Week  of April 5th.  Cryptography. Week  of March 22nd. Modular arithmetic and Cryptography  Project 2.   Due  April 5th. Week of March 15th, Modular arithmetic and Cryptography  Homework 5: Problems 16 and  and  this problem , due  March 22th This problem, Week of March 8th, Cryptography Keep working on  Project 2.     Homework 4: Problems 14, and 15 , due  March 15th Week of March 1st, start working on  Project 2.  Week of Feb 22th. Curve Fitting - Cryptography. Project 1, due  March 4st. The data is here. Week of Feb15th: Curve Fitting. Homework 3: Problems 9, 10, 12; due  Feb  25nd Week of Feb 8th:  Curve Fitting. Homework 2: Problems 6, 7, 8; due Thursday Feb  18th Week of Feb 1nd: Preliminaries about Mapple. Curve Fitting. Homework 1: Problems 1, 2 and 3; due Monday Feb 8th.  Week of Jan 25th: Preliminaries about Maple. If you have a computer, get the Maple CD from Seawolvers Market or Information Technology and install it. Also, submit Homework 0, due Feb 1st These are the topics to be covered during the term: Preliminaries about Maple, Curve Fitting, Cryptography and a third topic (which will be chosen later in the term.)