bigOmega_factorial_two
plain-language theorem explainer
The theorem asserts that the total number of prime factors counted with multiplicity for 2! equals 1. Number theorists using arithmetic functions to check small factorial cases would cite this when validating base computations before larger prime-counting arguments. The proof is a direct native_decide tactic that evaluates the equality by computing the factorial and its factorization.
Claim. Let $Ω(n)$ be the arithmetic function that counts the total number of prime factors of $n$ with multiplicity. Then $Ω(2!) = 1$.
background
The module supplies lightweight wrappers around Mathlib arithmetic functions, beginning with the Möbius function and extending to related maps such as bigOmega. bigOmega is defined as the arithmetic function cardFactors, which returns the total number of prime factors counted with multiplicity. The local setting keeps statements minimal to stabilize basic interfaces before layering Dirichlet inversion or deeper algebra.
proof idea
The proof is a one-line wrapper that applies the native_decide tactic to evaluate bigOmega (Nat.factorial 2) directly against the constant 1.
why it matters
This supplies an explicit small-case check inside the arithmetic-functions module that supports prime-factorization bookkeeping. It sits upstream of any future Möbius or inversion results in the same file and aligns with the Recognition Science practice of verifying concrete values on the phi-ladder before invoking the full forcing chain. No downstream theorems currently reference it.
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