# Spring 2009 MATH 302: Introduction to proofs via number theory Branko Ćurgus

Monday, June 8, 2009

• This is what I wrote during the exam.

Friday, June 5, 2009

• Here is my proof that among any sixteen consecutive integers there is always at least one integer relatively prime to the other fifteen.

• The usual update of the list of done and to-do problems.

Monday, June 1, 2009

Friday, May 29, 2009

• This is what I wrote during the exam.

Thursday, May 28, 2009
• The usual update of the list of done and to-do problems.

Tuesday, May 26, 2009
• The usual update of the list of done and to-do problems.

• Here I finish the proof that I started in class today.

Thursday, May 21, 2009
• The usual update of the list of done and to-do problems.

• Here is the second Assignment.

Monday, May 18, 2009
• The usual update of the list of done and to-do problems.

• Solutions for Assignment 1.

Thursday, May 14, 2009
• Chapter 3 of the class notes.

• The usual update of the list of done and to-do problems.

Monday, May 11, 2009

• The usual update of the list of done and to-do problems.

• The following statement we used often: Let u and v be positive integers. If uv = 1, then u = 1 and v = 1. Here is a proof. It is easier to prove the contrapositive: Let u and v be positive integers. If u > 1 or v > 1, then uv > 1. A proof of the contrapositive follows. Let u and v be positive integers such that u > 1 or v > 1. There are two cases. Case 1: u > 1 and v > 0, and Case 2: v > 1 and u > 0. In Case 1, Axioms 13 and 9 we have uv > v. Since v >= 1, we conclude uv > 1. In Case 2 Axioms 13 and 9 imply uv > u, and since u >=1, we again conclude uv > 1. The contrapositive is proved.

Thursday, May 7, 2009

• The usual update of the list of done and to-do problems.

• Here is the first Assignment.

Tuesday, May 5, 2009

• I am posting Grayce's proof of Proposition 1.4.1.

• The usual update of the list of done and to-do problems.

• Here are few statements (Lemmas, Propositions and Theorems, with proofs) related to Section 2.3.

Thursday, April 30, 2009
• The usual update of the list of done and to-do problems.

• Here are colored and dramatized proofs of Propositions 2.1.5, 2.1.7 and 2.1.10. I don't recall that we did 2.1.7 in class. The proof of 2.1.10 here is different from the proof that was done in class.

• Today I assigned a homework due on Monday. The homework is to write a complete proof of Proposition 1.4.1 in your own words and in your own style. You should study the proof given in class and the proof in the notes until you internalize the basic ideas. Then you can write a proof in your own words. I will grade these proofs by an integer between 1 and 10. These points will be added to your score on the exam. On Monday, please hand in your exam together with the homework.

Wednesday, April 29, 2009

• Dan gave a perfect solution for Problem 4 on Exam 1.

Tuesday, April 28, 2009

• This is what I wrote during the exam.

Monday, April 27, 2009
• Britt asked an interesting question today. This is a summary of my answer.

• The current list of done and to-do problems.

Friday, April 17, 2009

Thursday, April 16, 2009
• The list of done and to-do problems.

Tuesday, April 14, 2009
• The list of done and to-do problems.

Monday, April 13, 2009
• The list of done and to-do problems.

• A few thoughts about proofs.

• Here I collected 12 different ways to say: If p, then q.

Friday, April 10, 2009

Tuesday, March 31, 2009