The set of primes is supernatural: a Lean formalization of the statement of the conjecture

Abstract

The paper \emph{Conjecture: the set of prime numbers is supernatural} conjectures that no non-constant function built from the identity and constants by finitely many pointwise additions, multiplications, and exponentiations maps every positive integer to a prime. We give a complete Lean~4 formalization of that paper over Mathlib: every definition, example, remark, numbered result, and experimental table row has a machine-checked counterpart, with no \lcode{sorry}. The conjecture and similar generalizations are stated exactly, as named open problems. So stated, the conjecture becomes a precise target: an automated reasoning system can now attempt a kernel-checked proof.