IndisputableMonolith.NumberTheory.Primes.GCDLCM
The GCDLCM module supplies identities connecting greatest common divisors and least common multiples of natural numbers. Researchers handling multiplicative properties in the prime layer of Recognition Science would cite these results. The identities follow from direct algebraic reduction using standard divisibility rules.
claimIf $a$ and $b$ are coprime then $lcm(a,b)=a b$.
background
The module sits in the NumberTheory.Primes namespace and imports Mathlib together with the Basic primes module. The upstream Basic module supplies axiom-free prime footholds by reusing Mathlib's Nat.Prime library, with the explicit design goal to remain axiom-free and sorry-free for integration into the main monolith. GCDLCM adds the coprimeness identities required for algebraic simplifications of natural-number products.
proof idea
The module consists of direct applications of Mathlib gcd and lcm properties. Each result reduces algebraically from the coprimeness hypothesis without further tactics or auxiliary lemmas.
why it matters in Recognition Science
This module feeds the Primes namespace aggregator, which downstream code imports for all prime-related work. It supplies the coprime LCM product rule that supports higher algebraic manipulations in the number theory layer of the Recognition framework.
scope and limits
- Does not extend the identities to integers or other rings.
- Does not prove uniqueness of prime factorization.
- Does not address analytic estimates or density results.