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arXiv · 2604.09808

A formal proof of the Ramanujan--Nagell theorem in Lean 4

Barinder S. Banwait
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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.

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Barinder S. Banwait. 2026-04-10. A formal proof of the Ramanujan--Nagell theorem in Lean 4. https://arxiv.org/abs/2604.09808

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