2011 has been declared as a Number Theory Year at IMSc and it also coincides with Prof R Balasubramaniam’s 60th birthday. On that occasion, we have introductory courses on special topics in Number Theory by various imminent Number Theorists across India as well as abroad. This week, rather this fortnight, we have Pietro Corvaja from Italy speaking on Integral points on varieties: an introduction to Diophantine Geometry. Like most talks, I lost track of the seminar after about a couple of minutes. But till then, I witnessed some fantastic number theory, that I describe below:

Theorem: (Dirichlet – 1842) Let $\alpha, Q\in \mathbb R$ with $Q>1$. Then there are integers $p, q$ such that $1 \leq q < Q$ and

$| \alpha q - p | \leq \displaystyle \frac{1}{Q}$.

Proof: First consider the case when $Q$ is an integer. Then, $0, 1, [\alpha], [2 \alpha] , \cdots, [(Q-1)\alpha]$ are $Q+1$ numbers inside $[0,1]$. Partitioning the unit interval into $Q$ subintervals, namely, $[0, 1/Q), [1/Q, 2/Q), \cdots , [\frac{1}{(Q-1)}, 1]$ it follows from Dirichlet’s box principle that some interval must contain two of these points. Hence there must be nonnegative integers $p_1, p_2, q_1, q_2$ each less than $Q$ such $r_1 \neq r_2$ and

$|( r_1 \alpha - s_1) - (r_2 \alpha - s_2) | \leq \frac{1}{Q}$.

Since $r_1 \neq r_2$, the theorem follows for $Q\in\mathbb Z$.

For $Q\not\in\mathbb Z$, take $Q' = [Q] + 1$ and since the inequality holds with $\frac{1}{Q'}$; it must also hold when $Q'$ is replaced by $Q$.

Dirichlet was a brilliant mathematician who produced astonishing results by smartly applying the (Dirichlet’s) Box principle, more popularly known as the Pigeon-Hole principle. One of the best and useful theorems due to him is that an arithmetic progression with coprime first term and common difference must contain infinitely many primes. In the above theorem and its corollary, we see that irrational numbers can ‘nicely’ be approximated by rationals.

Corollary: Suppose that $\alpha$ is irrational. Then there are infinitely many rational numbers $\frac{p}{q}$ such that

$| \alpha - \frac{p}{q} | \leq \frac{1}{q^2}$

Proof: Obviously, we can demand $p, q$ to be relatively prime in the above theorem. Since $p\alpha - q$ is never zero, the inequality $| \alpha - \frac{p}{q} | \leq \frac{1}{Q}$ can be satisfied by finitely many $Q$‘s. Thus as $Q$ increases, the primes $p, q$ must also vary.

Note that the corollary is false if $\alpha$ is rational. There is a result due to Liouville which says that an algebraic number cannot be so ‘nicely’ approximated by rationals. This theorem due to Liouville also gives the first ever found transcendental number, the Liouville number. Returning back, Roth said that Dirichlet’s bound could not be improved upon using the same hypotheses. Here is the precise statement (without proof) :

Theorem: (Roth) If $\alpha \in \overline{\mathbb Q} \cap \mathbb R$ and $\varepsilon >0$, then there are only finitely many rationals satisfying $|\alpha - \frac{p}{q}| < \frac{1}{q^{2+\varepsilon}}$.

Another fascinating result due to Thue follows below from Roth’s theorem. (There was a lot of mathematics in the talk that I am not mature enough to follow. Nevertheless, the following theorem needed none of the previous discussion in its proof).

Theorem: (Thue) If $F(X,Y)\in\mathbb Z[X,Y]$ is a homogeneous polynomial of degree at least 3 and if it has distinct roots over $\latex \overline{\mathbb Q}$, then for every integer $c$, the equation

$F(x,y) = c$

has only finitely many solutions $(x,y)\in\mathbb Z^2$.

Proof: Since $F$ is homogeneous, we can write $F(X,Y) = a (X - \alpha_1 Y) \cdots (X - \alpha_n Y)$ with all the $\alpha_i \in\overline{\mathbb Q}$‘s distinct. Then, if $(x,y)\in\mathbb Z^2$ is a solution,

$a (\frac{x}{y} - \alpha_1) \cdots (\frac{x}{y} - \alpha_n) = \frac{c}{y^n} \rightarrow 0$ as $y \rightarrow \infty$.

Thus, at least one of the terms on the left, which we assume WLOG to be the first, goes to zero. This means that there is a $\delta>0$ such that $|\frac{x}{y} - \alpha_i | > \delta$ for $i = 2, 3, \cdots, n$.

Now, $\displaystyle|\frac{a}{c^n}|. \frac{1}{y^n} = | \frac{x}{y} - \alpha_1 | \prod_{j=2}^n | \frac{x}{y} - \alpha_j | \geq | \frac{x}{y} - \alpha_1 | . \delta^{n-1}$. Thus, $|\frac{x}{y} - \alpha_1 | \leq | \frac{c}{a\delta^{n-1}}| . \frac{1}{y^n}$. If there are infinitely many solutions $(x,y)\in\mathbb Z^2$, then $y$ can be made as large as possible. But this contradicts Roth’s theorem if $n>2$.

Thus, we can say, $X^3 + 3 Y^3 = 1$ can have at most finitely many integer-valued solutions. This is a deep enough result to be interesting, isn’t it?