Math 645: A Survey of Mathematical Problems I
Frank Sottile

• The homework this week is more graph theory; I think a bit easier than last week's, as that included proving an inequality (inequalities in discrete mathematics are a bit harder than those in analysis, as the manner of thinking in discrete mathematics is antithetical to inequalities). You will find drawing K₆ and K₇ on the torus to be a bit challenging.
  The proof of the four color theorem was about 45 years ago, by Appel and Haken of the University of Illinois. It was somewhat controversial; many mathematicians and philosophers of mathematics refused to accept it as it used a computer to both generate a large list of unavoidable configurations, and then to apply the algorithmic procedure of discharging to check that each one may be colored when inserted back into the graph. In the 1990's a different proof was conceived and exectuted—while that second proof still required a computer to check it, parts of it may be checked by hand, and it is somewhat simpler. With two proofs, and many more eyes having looked at this, I think that the controversy has receeded.
• Our reading starts us down the path of number theory. It is a grand topic, and the piece by Richards sets the stage. You should note that that piece is now dated. Fermat's Last Theorem is now Wiles' Theorem (proved in the 1990's), but the Riemann Conjecture is still open. However, the computational evidence for it is amazing (I am not up on that part of the literature, but perhaps we now know that the first 30 million zeroes all have real part 1/2.) Catalan's Conjecture was also solved since the article was written, and Falting's Field medal (1986) was for a generalization of Thue's result about bivariate Diophantine equations.
  Peterson's piece is a fun romp through some number sequences. The ultimate source of number sequences is the OEIS.

• Read Selections 15 (by Ian Richards) and 16 (by Ivars Peterson), which will help us get into number theory.
• Chapter 3 of A Survey of Mathematical Problems.
• Dr. Geller's Lecture 7.

1. Exercises 3.11–3.13, and Problem 3.11. There is an error in Problem 3.11 in the hint about K₅. It is more complicated than not containing a K₅. You will want to re-read the reading on the history of the four colour theorem.
2. Extend Problem 3.10, looking for a drawing of K₆ on a torus, and perhaps K₇. One way to help with this is to think of a torus as two congruent annuli (an annulus is a circle with a circle cut out) attached together along the corresponding circles; think of this as viewing a donut from above. Another way is a one annulus with the inside and outside circles glued together.