mobius_thirtyfive
μ(35) equals 1 because 35 factors as 5 times 7 and is squarefree with exactly two distinct primes. Analytic number theorists cite this explicit evaluation when checking small cases in Möbius inversion or Dirichlet convolution. The proof reduces to a single native decision step that evaluates the arithmetic function definition directly.
claimThe Möbius function satisfies $μ(35) = 1$.
background
The module supplies lightweight wrappers around Mathlib's arithmetic functions, beginning with the Möbius function μ defined as an arithmetic function from naturals to integers. The upstream definition identifies mobius as the standard Möbius function ArithmeticFunction.moebius. This specific theorem records the value at 35, which factors as 5 times 7 and is squarefree with two distinct prime factors.
proof idea
The proof is a one-line wrapper that invokes native_decide to compute the value of the Möbius function at 35 from its definition.
why it matters in Recognition Science
This value contributes to the collection of explicit Möbius evaluations in the arithmetic functions module, supporting downstream calculations in number theory such as inclusion-exclusion or Dirichlet convolution. It aligns with the general property that μ(n) equals (-1)^k for squarefree n with k prime factors. No immediate parent theorems are listed, but it populates the basic interface for the Möbius function in the Recognition Science number theory layer.
scope and limits
- Does not prove the general formula for μ at arbitrary squarefree integers.
- Does not address non-squarefree cases where μ vanishes.
- Does not connect to physical constants or forcing chains in the broader framework.
formal statement (Lean)
1110theorem mobius_thirtyfive : mobius 35 = 1 := by native_decide
proof body
Term-mode proof.
1111
1112/-- μ(42) = -1 (squarefree with 3 prime factors). -/