bigOmega_eighthundredforty
plain-language theorem explainer
Ω(840) evaluates to 6 under the total prime factor counting function. Researchers applying arithmetic functions in number theory would cite this specific case for verification of the cardFactors implementation. The verification relies on a native decision procedure that directly computes the factorization sum.
Claim. The total number of prime factors of 840, counted with multiplicity, is 6: $Ω(840) = 6$.
background
This module supplies lightweight wrappers around Mathlib's arithmetic function library, beginning with the Möbius function and extending to related functions such as the total prime factor count. The bigOmega abbreviation is defined as ArithmeticFunction.cardFactors, which implements Ω(n) as the sum of the exponents in the prime factorization of n.
proof idea
The proof is a one-line term that invokes the native_decide tactic. This tactic evaluates the arithmetic expression by computing the prime factorization of 840 internally and summing the exponents to obtain 6.
why it matters
This declaration provides a concrete numerical check for the bigOmega function within the arithmetic functions module. It supports the broader development of number-theoretic tools in the Recognition framework, though it does not yet feed into downstream theorems. The module focuses on Möbius footholds as a starting point for deeper inversion formulas.
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