Graduate Course in Number Theory

I just completed my second graduate math course. The subject was number theory which is known as the "queen of mathematics." Before I describe the course I want to briefly mention that last summer I toured the old growth redwood forests in California. When I walked through the redwoods I felt like I was in an enchanted forest. I marveled at the towering sequoias which ranged in age from 1,500 to 300 years old. Each tree is a monument to nature herself and the forest has an eternal quality to it.

I found my number theory course was similar to visiting the ancient forest. We learned theorems from Euclid (e.g. the infinitude of prime numbers) and Diophantus that date back over 2,000 years. We learned venerable theorems from Fermat, Euler, and Gauss that date back over 300 years. Each theorem stands as a pillar of mathematical truth for all eternity.

I have to admit, though, that I was lost in the forest of number theory at times. At certain points I could understand individual theorems and their proofs, but for long stretches I stumbled along. The journey began with chapters on divisibility and congruences. These are straightforward topics that can be taught to bright high school students or the brightest middle school students. Fermat’s Little Theorem and Euler’s generalization were covered. Of course, the totient function f(n) was defined (the count of numbers relatively prime to n) and in a later chapter the remarkable formula was proven. Translated into English this states that the sum of the totient function ranging over all of the divisors of n is equal to n itself (futher translation not available).

Okay they were the easy chapters. Now we move onto quadratic reciprocity and quadratic forms. Our textbook An Introduction to the Theory of Numbers by Niven, Zuckerman, and Montgomery mentions that "Gauss discovered the quadratic reciprocity law just before his 18^th birthday. After a year of strenuous effort he found the first proof, in 1795, at the age of nineteen." I find this fact comforting. The world’s greatest mathematician had to struggle for a year to find a proof for a fact that he knew to be true. So it’s okay to struggle with math homework – that’s its purpose in life.

Binary quadratic forms have the form f(x,y) = ax²+bxy + cy². Mathematicians have studied BQFs intensely and have a complete understanding of them. I can’t say that I have a complete understanding but I did learn a technique based on BQFs that can be used for the following arithmetic parlor trick. Given the prime number 398417 find two numbers whose squares sum to 398417 (answer: 631² + 16²).

For some reason our teacher decided to skip Chapter 4 which has marvelous results like the Moebius inversion formula (don’t ask). So we continue with the chapter on Diophantine equations, the most famous of which is Fermat’s Last Theorem (there are no integer solutions for xⁿ+ yⁿ = zⁿ where n > 2). This leads to an area of mathematics I had never learned before called elliptic curves (these are not elliptical curves). The geometric analysis of these curves (chord and tangent method) yields additional solutions to the Diophantine equation on which they are based. In our class we visit the foothills of the mountains that Andrew Wiles conquered in his 1993 proof of Fermat’s Last Theorem.

We skip chapter 6 to get to the chapter on Continued Fractions. I’m in the embarrassing situation of having to tell people that I’m studying fractions in graduate mathematics, but it’s worth it. These fractions are small wonders and the basis for new insights into irrational numbers. I also learned of Pell’s equation and was able to fully understand an anecdote relating to the Indian mathematician Ramanujan. One of his friend’s gave him a puzzle to determine a house number that met certain conditions. Ramanujan was cooking at the time, but he was able to instantly state the result in general terms using a continued fraction based on Pell’s equation.

For the last two classes our teacher covered eclectic topics in multiplicative number theory. At the very least I learned what Dirichlet’s series are and the Reimann zeta function in particular. Oh to solve the Riemann hypothesis relating to the roots of the zeta function!

Note on textbook: It seems that the textbook by Niven, Zuckerman andMontgomery is widely used as a graduate text. It has survived the test of time having endured five editions since 1961. My belief is that it was a fit text in its early editions, but by the fifth edition it has grown flaccid and overweight. The authors give the most succinct proofs and even skip steps at times.

If I felt stultified by the topics at hand, the authors helped me achieve that state.