🤖 AI Summary
This work investigates whether there exists a non-constant function, constructed from the identity function and constants through finitely many pointwise additions, multiplications, and exponentiations, that maps every positive integer to a prime number. The author formally articulates—within the Lean 4 proof assistant using the Mathlib library—the conjecture that the set of primes is "supernatural," along with its natural generalizations, as precisely stated open problems. The entire development, including definitions, theorems, examples, and computational experiments, has been fully formalized and verified by the Lean kernel without reliance on any unproven assumptions. This provides a rigorous, machine-checkable foundation for future automated reasoning and formal proofs concerning this 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.