RefacteThe mechanism
Build an exact arithmetic selector
Turn a test for primality into arithmetic that selects the first prime above a bound. Factorial congruences, greatest common divisors, and valuations organise the selection. The construction is useful here because each stage has an exact role in isolating the answer.
What becomes elementary: The desired selection is expressed by a finite formula, with its correctness separated from questions of computational efficiency.
Track what each arithmetic test retains. The study is an exact construction; it does not claim a practical improvement over prime-search algorithms.
For another construction, explore the mechanism calculi. Edition dates, sources and reference chapters identify the scope of each study.
The construction turns tests for primality into a formula that isolates the first prime beyond a prescribed integer. The point is the exact arithmetic mechanism, rather than a practical prime-search algorithm.
Background
Elementary number theory, factorials, and divisibility.
Read the full exposition
The complete argument is available as a PDF.
Revision history ·
Revised October 7, 2026: clarified the introduction while preserving the problem’s IRC provenance. The theorem, proof, and appendix are unchanged.