three_almost_prime_eighteen
plain-language theorem explainer
The statement confirms that 18 factors as 2 times 3 squared and therefore carries exactly three prime factors counted with multiplicity. Number theorists checking small cases inside the arithmetic functions library would reference this instance to validate the bigOmega predicate. The proof reduces to a single native_decide step that evaluates the equality directly from the definition.
Claim. $Ω(18) = 3$, where $Ω(n)$ denotes the total number of prime factors of $n$ counted with multiplicity.
background
The module supplies lightweight wrappers around Mathlib arithmetic functions, beginning with the Möbius function and extending to bigOmega. A number is defined to be 3-almost prime precisely when its total prime factor count with multiplicity equals three. Upstream results include the definition isThreeAlmostPrime (n : ℕ) : Bool := bigOmega n = 3 together with interface declarations such as collision-free structures from OptionAEmpiricalProgram and algebraic tautologies from SimplicialLedger.
proof idea
The proof is a one-line wrapper that applies native_decide to discharge the equality isThreeAlmostPrime 18 = true by direct computation of bigOmega 18.
why it matters
This concrete verification anchors the 3-almost-prime predicate in the arithmetic functions library. It supports downstream checks in prime-related number theory within the Recognition framework, though no immediate parent theorems are listed. The result aligns with the need for explicit small-case validations before layering Dirichlet algebra.
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