In 1978, Roger Apéry proved that the number ζ(3) — the sum of 1/n³ for all positive integers n — is irrational. The proof was a miracle. Not metaphorically: mathematicians literally used that word. The proof worked by constructing a specific sequence of rational approximations to ζ(3) that converged too fast to be consistent with rationality. The construction appeared out of nowhere, with no clear motivation, and no one could explain why it worked or how to generalize it.
For 47 years, ζ(3) remained the only odd zeta value proven irrational by this method. The trick seemed unrepeatable — a one-time insight that illuminated a single number without revealing anything about the landscape around it.
Frank Calegari, Vesselin Dimitrov, and Yunqing Tang changed this by finding the framework behind the miracle. They showed that Apéry's sequence wasn't an isolated construction but an instance of a general method: certain power series, when analyzed through the lens of complex algebraic geometry, automatically produce irrationality proofs for the values they encode. The method generates Apéry-like sequences systematically, not by guesswork.
The distinction between a trick and a method is structural. A trick solves one problem. A method solves a class of problems by identifying the shared structure that makes them tractable. Apéry's proof was a trick — brilliant, irreproducible, opaque. Calegari-Dimitrov-Tang's framework is a method — it explains why the trick worked and produces new tricks for new problems. They proved new irrationality results for values related to modular forms, results that would have been unreachable by Apéry's techniques alone.
The framework uses the arithmetic of algebraic varieties — geometric objects defined by polynomial equations over number fields. The connection between power series and arithmetic geometry runs through a classical object called a period, which encodes geometric information as a number. Periods of algebraic varieties include values like π, log 2, and ζ(3). The irrationality of these values is connected to the geometry of the varieties that produce them, and the framework makes this connection explicit and computable.
What makes this resolution satisfying is not just that it works but that it explains the mystery. For 47 years, Apéry's proof was magical — it worked, and no one knew why. Now the magic has a source. The power series that Apéry chose wasn't arbitrary; it was the unique series whose complex-analytic properties forced rapid rational approximation. The construction that seemed unmotivated was, in retrospect, the only construction that could have worked, given the geometric constraints of the underlying variety. The miracle was inevitable. It just took 47 years to understand why.