Math 645: A Survey of Mathematical Problems I
Frank Sottile

• I am having a particularly tough week, as I've had to write seven professional letters (promotions, national prizes, scholarships for students, etc) this week, and I am getting ready to go back to Providence, RI on Wednesday. Consequently, my commentary is a bit short this week. I expect that everything will be better next week.
• The reading is short, but a bit hard. Fermat is discussing his "little Theorem" in examples. Here is some of the mathematics going on. 3³-1 is a multiple of 13, which implies that 3⁶-1 is a multiple of 13. This is because 3⁶-1=(3³-1)(3³+1). The same reasoning gives that 3¹²-1 is a multiple of 13, which is an instance of Fermat's little theorem.
      Note how hard it is to understand what he is doing; he has a very different way of expressing himself, and does not use algebraic notation. This topic, perfect numbers, and Pell's equation are all discussed in the course Math 629: History of Mathematics.
• Dr Geller's Lecture 10 contains some hints for the problems. I expect to discuss a lot of this on Piazza with you. Start early!

• Read Selection 23, it is short.
• Chapter 4 of A Survey of Mathematical Problems.
• Dr. Geller's Lecture 10.

1. Exercises 4.10 4.12 and Problems 4.9 4.12.
2. Please let me know your topic for the term paper.