practical_thirtytwo
plain-language theorem explainer
The sum-of-divisors function satisfies σ_1(32) ≥ 32, confirming the divisor-sum threshold for 32 in this arithmetic setting. Number theorists working with practical numbers or lightweight wrappers around Mathlib's sigma function would cite the instance. The proof is a direct native decision that evaluates the concrete sum of divisors of 32.
Claim. $σ_1(32) ≥ 32$
background
The module supplies small wrappers around Mathlib arithmetic functions, beginning with the Möbius function μ and extending to the sum-of-divisors function σ_k. The local setting keeps statements lightweight, deferring deeper Dirichlet algebra and inversion until basic interfaces stabilize. The relevant upstream definition is the abbrev sigma (k : ℕ) : ArithmeticFunction ℕ := ArithmeticFunction.sigma k, which supplies the operator σ_1 used in the inequality.
proof idea
The proof is a one-line wrapper that applies native_decide to evaluate the concrete numerical inequality σ_1(32) ≥ 32 by direct computation of the divisor sum.
why it matters
This supplies a verified base case for the divisor-sum condition on practical numbers inside the NumberTheory.Primes.ArithmeticFunctions layer. It supports the module's role as a foothold for Möbius-related tools without feeding any downstream theorems in the current graph. The check aligns with the framework's pattern of concrete verifications that can later interface with the main forcing chain or RCL identities.
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