three_almost_prime_twentyeight
plain-language theorem explainer
28 satisfies the 3-almost-prime condition because its prime factorization 2² × 7 has exactly three prime factors counted with multiplicity. Researchers working on arithmetic functions or explicit prime-omega checks inside the Recognition Science number-theory layer would cite this verified instance. The proof is a one-line native decision that directly evaluates the bigOmega definition.
Claim. $Ω(28) = 3$, where $Ω(n)$ denotes the total number of prime factors of $n$ counted with multiplicity.
background
The module supplies lightweight wrappers around Mathlib's arithmetic-function library, beginning with the Möbius function μ. A number n is 3-almost prime precisely when bigOmega n equals 3. The upstream definition states: A number is 3-almost prime if Ω(n) = 3. Sibling results establish basic properties of bigOmega for squarefree numbers and prime powers.
proof idea
The proof is a one-line wrapper that invokes native_decide to evaluate the Boolean equality isThreeAlmostPrime 28 = true by direct computation of bigOmega 28.
why it matters
This concrete verification supports the arithmetic-functions layer that supplies Möbius footholds for later Dirichlet algebra. It forms part of the explicit checks that stabilize the primes module before deeper inversion formulas are added. No downstream uses are recorded, so the result remains a local instance that can be referenced when prime-factor counts appear in mass-ladder or phi-ladder constructions.
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