Deep Vision: A Formal Proof of Wolstenholmes Theorem in Lean 4

Abstract

We present a formal verification of Wolstenholme's theorem -- $\binom{2p}{p} \equiv 2 \pmod{p^3}$ for prime $p \geq 5$ -- in Lean~4 with Mathlib. The proof proceeds by expanding the shifted factorial product $\prod_{k=1}^{p-1}(p+k)$ to second order in $p$, identifying the quadratic coefficient as the second elementary symmetric product, and showing its divisibility by $p$ via power sum vanishing in $\mathbb{Z}/p\mathbb{Z}$. The formalization comprises nine lemmas across approximately 800 lines of Lean, with zero \texttt{sorry} declarations. To our knowledge, this is the first formal verification of Wolstenholme's theorem in Lean~4. The proof was discovered through a collaboration between a relational analogy engine for theorem proving and human-directed formalization.