A formal proof of the Ramanujan--Nagell theorem in Lean 4
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913178606108672 |
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| author | Banwait, Barinder S. |
| author_facet | Banwait, Barinder S. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_09808 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A formal proof of the Ramanujan--Nagell theorem in Lean 4 Banwait, Barinder S. Number Theory Logic in Computer Science 11D61, 68V20 (primary), 11R04 (secondary) 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. |
| title | A formal proof of the Ramanujan--Nagell theorem in Lean 4 |
| topic | Number Theory Logic in Computer Science 11D61, 68V20 (primary), 11R04 (secondary) |
| url | https://arxiv.org/abs/2604.09808 |