🤖 AI Summary
This work presents a complete formalization of the Ramanujan–Nagell theorem in Lean 4, leveraging the Mathlib library to establish that the Diophantine equation $x^2 + 7 = 2^n$ admits exactly five integer solutions: $(n, x) = (3, \pm1), (4, \pm3), (5, \pm5), (7, \pm11), (15, \pm181)$. The proof is carried out within an interactive theorem prover by developing foundational structures from algebraic number theory, including the ring of integers, class group, and unit group of the quadratic field $\mathbb{Q}(\sqrt{-7})$. This effort not only provides the first machine-verified proof of the theorem but also significantly extends the reusable infrastructure for algebraic number theory in formalized mathematics.
📝 Abstract
We present a complete formalization, in the Lean interactive theorem prover with the Mathlib library, of the Ramanujan--Nagell theorem: the only integer solutions to the Diophantine equation $x^2 + 7 = 2^n$ are $(n,x) \in \{(3,\pm1),(4,\pm3),(5,\pm5),(7,\pm11),(15,\pm181)\}$. The formalization includes all dependencies, notably the computation of the ring of integers of the quadratic field $\mathbb{Q}(\sqrt{-7})$, its class number, and unit group. We describe the proof strategy, the architecture of the formalization, and the challenges encountered in bridging the gap between textbook proofs and their machine-checked counterparts, with particular attention to the algebraic number theory infrastructure required.