Math 645: A Survey of Mathematical Problems I
Frank Sottile

• We have two fairly easy readings this week, numbers 20 and 21 in the source book. The proof that π is irrational should be done in Calc II (I do it when I teach that class). I assigned reading 22, but only read the first six pages (on irrationality and trigonometric functions), and the proof of Theorem 2.12, that e is transcendental.
• Here are some hints: Geller suggests proving Fermat's little theorem in the easier form a^p ≡ a (mod p). Use induction on a, first proving the Freshman's dream (a+b)^p = a^p + b^p (mod p).
  I hope that all of you know the telescoping factorization aⁿ + bⁿ = (a + b)(a^n-1 - a^n-2b + ... + b^n-1), at least when n is odd. Also, you all know how to factor a difference of squares....

• Read Selections 20, 21, and 22 (22 is more involved).
• Chapter 4 of A Survey of Mathematical Problems.
• Dr. Geller's Lecture 9, which is a paragraph.

1. Exercises 4.7, 4.8, and 4.9, and Problems 4.5, 4.6, 4.7, and 4.8. (The right way to understand 4.8 is given in a hint above. It leads to an interesting factorization of x⁶+ 27. Can you discover this?
2. Start to work on your term papers.