six_almost_prime_sevenhundredtwentynine
plain-language theorem explainer
729 equals 3 to the sixth power and therefore carries exactly six prime factors counted with multiplicity. Researchers verifying concrete instances of the bigOmega arithmetic function would cite this specific check. The proof reduces to a native decision procedure that evaluates the boolean predicate directly on the constant input.
Claim. $Ω(729) = 6$, where $Ω(n)$ counts the total number of prime factors of $n$ with multiplicity.
background
The module supplies lightweight wrappers around Mathlib arithmetic functions, beginning with the Möbius function and extending to bigOmega. A number $n$ is 6-almost-prime precisely when bigOmega $n$ equals 6, as stated in the sibling definition isSixAlmostPrime. The local setting keeps statements lightweight to support later Dirichlet algebra and inversion once basic interfaces stabilize.
proof idea
The proof is a one-line wrapper that invokes native_decide to evaluate the boolean predicate isSixAlmostPrime 729 by direct computation of bigOmega 729.
why it matters
This concrete verification anchors the arithmetic function library at the value 729 = 3^6 inside the NumberTheory.Primes.ArithmeticFunctions file. It supplies a checked case for prime factorization counts that can feed downstream prime-related constructions in the Recognition Science framework. No parent theorems appear in the used-by graph.
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