Math 645: A Survey of Mathematical Problems I
Frank Sottile

• This week we start on number theory, which Gauß called "The Queen of Mathematics". Our first reading by Victor Klee and Stan Wagon is a bit odd, as it repeats itself a bit (Chapters 14 and 15 occur twice). To me it appears these are from different editions of the book, with the second appearance deeper and more complete. A number of interesting gems are mentioned in this part; Egyptian fractions show that the algorithms we use in arithmetic are not the only ones, nor are our methods of representing numbers the only way. This reappears in courses on the History of Mathematics (Math 629, which I have taught before).
      One of your homework problems asks you to look up an update to this reading. For others, look up "arithmetic sequences of primes", and "twin primes". There is an interesting video of Terry Tao on the Colbert Report related to this.
• The second reading is an article that appeared in the American Mathematical Monthly, or Monthly, which is the more widely-read journal/magazine devoted to mathematics (I am currently one of its editors). It assumes some background, which I will try to sketch for you. A number is algebraic if it is a root of a polynomial equation with integer coefficients; otherwise it is transcendental. There is a long history of asking if numbers such as π or e are algebraic or transcendental; the ancient Greek's problem of "Squaring the circle" is answered definitively in the negative by showing that π is transcendental. In the first half of the 19th century, the French Mathematician Liouville showed that there are many transcendental numbers. For example, his argument shows that the obvious number that begins 0.101001000100001000001000000100000001000000001000... is transcendental. Georg Cantor literally "blew this out of the water" by showing that most real numbers, in a very strong sense, are transcendental, and this article reviews that while letting the author grind his axe that Cantor's proof is constructive.
      All mathematicians should be aware of Cantor's proof; we teach it in Math 220 and again in Math 409. I first saw it in High School, when my calculus teacher, Carl Pasbjerg (his son comes up in a Google search), presented it to our class.
• Do appreciate Bloom's proof that the square root of 2 is irrational for its compactness.

• Read Selections 17 (Klee and Wagon), 18 (Robert Gray), and 19 (David M. Bloom) from our course packet.
• Chapter 4 of A Survey of Mathematical Problems.
• Dr. Geller's Lecture 8.

1. Look up Mersenne primes, and write a short paragraph on what you discovered. Look for answers to the following question: What is the largest Mersenne prime currently known, and how are they found?
2. Exercises 4.1–4.6 and Problems 4.1–4.4 from A Survey of Mathematical Problems.