three_almost_prime_fiftytwo
plain-language theorem explainer
The theorem verifies that 52 factors as 2 squared times 13 and therefore satisfies Ω(52) = 3. Number theorists assembling tables of small almost-primes within the Recognition Science arithmetic layer would cite this check. The proof reduces to a single native_decide evaluation of the bigOmega predicate.
Claim. $52 = 2^2 × 13$ is a 3-almost prime, i.e., the total number of prime factors counting multiplicity satisfies Ω(52) = 3.
background
The ArithmeticFunctions module supplies lightweight wrappers around Mathlib arithmetic functions, beginning with the Möbius function μ and the total prime-factor counting function bigOmega. A natural number n is 3-almost prime precisely when bigOmega n equals 3. The supplied definition isThreeAlmostPrime n := bigOmega n = 3 therefore translates the classical Ω predicate into a Bool for direct computation.
proof idea
The proof is a one-line wrapper that invokes native_decide on the closed term isThreeAlmostPrime 52, which reduces via the definition to the equality bigOmega 52 = 3 and evaluates to true by direct arithmetic.
why it matters
This supplies a concrete, machine-checked instance inside the primes arithmetic-functions module. It supports the module's stated purpose of furnishing Möbius footholds for later Dirichlet inversion work. No downstream theorems currently reference it, and it touches none of the core Recognition Science landmarks such as the J-function or the eight-tick octave.
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