A few years ago I discovered an amusing proof of Euclid’s theorem that there are infinitely many primes which I thought I’d record here for posterity. (I subsequently learned that a similar argument appears in this paper by Paul Pollack.)
To motivate the proof, and illustrate its working in an (admittedly silly) special case, suppose that there were just two prime numbers, called p and q. Then by the fundamental theorem of arithmetic (i.e., unique factorization into primes) we would have the following identity between generating functions:
Indeed, there is precisely one term for each integer on the left-hand side, and the same is true for the right-hand side (consider separately the cases where but , but , and ). Using the formula for the sum of a geometric series, we can rewrite our formula as an identity between complex-analytic functions valid on the open unit disc :
This is impossible, however, as we see by letting approach a primitive root of unity, since each term stays bounded except for , which tends to infinity.
My thesis advisor Robert Coleman passed away two years ago today (see this remembrance on my blog). One of the things I learned from Robert is that p-adic numbers have many unexpected applications (see, for example, this blog post). Today I want to share one of my favorite surprising applications of p-adic numbers, to a simple problem in Euclidean geometry. Continue reading
What is the probability that two randomly chosen integers have no prime factors in common? In honor of Pi Day, I’d like to explain the surprising answer: .
The hero of this story is Leonhard Euler, who worked out this astonishing connection between prime numbers and through a series of brilliant insights. In the spirit of Euler, I will be rather cavalier about issues of convergence and rigor here, focusing on the key underlying ideas.
Today marks the birthday of Karl Friedrich Gauss, the “Prince of Mathematicians”, who was born on April 30, 1777. In honor of Gauss’s 238th birthday, I thought I would blog about one of Gauss’s favorite theorems — the Law of Quadratic Reciprocity — and its relation to the sign of the quadratic Gauss sum, which we will determine using the Discrete Fourier Transform. Our exposition mostly follows this paper by Ram Murty. Regarding the sign of the quadratic Gauss sum, Gauss conjectured the correct answer in his diary in May 1801, but it took more than four years until he was able to find a proof in August 1805. Gauss wrote to his friend W. Olbers that seldom had a week passed for four years that he had not tried in vain to prove his conjecture. Then:
Finally, two days ago, I succeeded – not on account of my hard efforts, but by the grace of the Lord. Like a sudden flash of lightning, the riddle was solved. I am unable to say what was the conducting thread that connected what I previously knew with what made my success possible.
Ferdinand von Lindemann
In my last blog post, I discussed a simple proof of the fact that pi is irrational. That pi is in fact transcendental was first proved in 1882 by Ferdinand von Lindemann, who showed that if is a nonzero complex number and is algebraic, then must be transcendental. Since is algebraic, this suffices to establish the transcendence of (and setting it shows that is transcendental as well). Karl Weierstrass proved an important generalization of Lindemann’s theorem in 1885.
The proof by Lindemann that pi is transcendental is one of the crowning achievements of 19th century mathematics. In this post, I would like to explain a remarkable 20th century proof of the Lindemann-Weierstrass theorem due to Bezivin and Robba [Annals of Mathematics Vol. 129, No. 1 (Jan. 1989), pp. 151-160], which uses p-adic analysis in a key way. Their original argument was made substantially more elementary by Beukers in this paper; we refer the reader to [American Mathematical Monthly Vol. 97 Issue 3 (Mar. 1990), pp. 193-197] for a lovely exposition of the resulting proof, which rivals any of the usual approaches in its simplicity. But I’d like to focus here on the original Bezivin-Robba proof, which deserves to be much better known than it is. In the concluding remarks, we will briefly discuss a 21st century theorem of Bost and Chambert-Loir that situates the Bezivin-Robba approach within a much broader mathematical framework. Continue reading