semiprime_thirtyeight
plain-language theorem explainer
Verification that 38 factors as 2 times 19 establishes it as semiprime under the definition Omega(n) equals 2. Number theorists building tables of arithmetic functions in the Recognition framework cite this for explicit factorization checks. The proof is a one-line native_decide evaluation of the bigOmega predicate.
Claim. The integer 38 is semiprime, i.e., $38 = 2 × 19$ and thus $Ω(38) = 2$.
background
The module supplies lightweight wrappers around Mathlib arithmetic functions, beginning with the Möbius function μ. The predicate isSemiprime is defined by bigOmega n = 2, where bigOmega counts prime factors with multiplicity. Upstream results include the isSemiprime definition itself together with foundation structures that enforce collision-free or tautological properties in related modules.
proof idea
The proof is a one-line wrapper that invokes native_decide to evaluate the Boolean expression isSemiprime 38 directly from its definition bigOmega 38 = 2.
why it matters
This concrete instance populates the arithmetic functions library with verified semiprime examples. It supports the NumberTheory.Primes section but does not engage the Recognition Science forcing chain T0-T8 or constants such as phi. No downstream theorems are recorded yet.
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