three_almost_prime_twelve
plain-language theorem explainer
The declaration establishes that 12 factors as 2 squared times 3 and therefore satisfies the 3-almost-prime condition. Researchers tabulating small almost-primes for use in arithmetic-function identities would cite this verified instance. The proof reduces immediately to a native decision that computes the big-omega count on the concrete integer 12.
Claim. $Ω(12)=3$, 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. A number is defined to be 3-almost prime precisely when its total prime-factor count with multiplicity equals three. The upstream definition implements this check via the bigOmega function.
proof idea
The proof is a one-line wrapper that invokes native_decide to evaluate the boolean expression isThreeAlmostPrime 12.
why it matters
This concrete verification supplies a checked base case for the 3-almost-prime predicate inside the primes arithmetic-functions module. It supports downstream constructions that rely on explicit small almost-primes, although no immediate parent theorem is recorded. Within the broader framework the result anchors the arithmetic side of the number-theory layer that feeds into the forcing chain and phi-ladder mass formulas.
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