In honor of Pi Day 2023, I’d like to discuss Hilbert’s 7th Problem, which in an oversimplified (and rather vague) form asks: under what circumstances can a transcendental function take algebraic values at algebraic points?

The connection with is that Lindemann proved in 1882 that the transcendental function takes transcendental values at every nonzero algebraic number. Since by Euler’s formula, this proves that , and hence itself, is transcendental. In light of this theorem, it is natural to wonder what if anything is special here about the function and the point .

One thing that’s special about is that if is algebraic and is also algebraic, then both and are algebraic for all , and these numbers are all distinct. So one might be led to speculate that if is a transcendental entire function then there are only finitely many algebraic numbers for which is also algebraic.

Unfortunately, as Hilbert knew, this is completely false. For example, the function is transcendental but it takes the rational value 1 at every integer. In 1886, Weierstrass had given an example of a transcendental entire function that takes rational values at all rational numbers; later, in 1895, Stäckel showed that there is a transcendental entire function that takes rational values at all algebraic points. However, the functions of Weierstrass and Stäckel, are in some sense “pathological”; they have large growth rates and do not occur “in nature”. The challenge is to make this intuitive feeling more precise, and also to distinguish from .

One thing that is special about , which is not shared by any of the other functions mentioned in the previous paragraph, is that it satisfies a linear differential equation with rational coefficients (namely ). The existence of such a (not necessarily linear) differential equation turns out to be the key idea needed to generalize Lindemann’s theorem in a substantial way.

Another fruitful generalization is to rephrase our original question as an unlikely intersection problem: given two algebraically independent entire functions and satisfying suitable hypotheses, can we conclude that there are only finitely many complex numbers such that and are simultaneously algebraic? This generalizes our original question by letting and .

My goal in this post is to describe a surprising and beautiful method (the Stern-Brocot tree) for generating all positive reduced fractions. I’ll then discuss how properties of the tree yield a simple, direct proof of a famous result in Diophantine approximation due to Hurwitz. Finally, I’ll discuss how improvements to Hurwitz’s theorem led Markoff to define another tree with some mysterious (and partly conjectural) similarities to the Stern-Brocot tree.

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 →