sigma_three_five
plain-language theorem explainer
The equality σ_3(5) = 126 holds by direct evaluation of the sum of cubes over the divisors of 5. Number theorists checking small cases of the sum-of-divisors function would cite it as a concrete verification. Native_decide evaluates the arithmetic expression at compile time without manual expansion.
Claim. $σ_3(5) = 126$, where $σ_k(n)$ denotes the sum of the $k$th powers of the positive divisors of $n$.
background
The module supplies lightweight wrappers around Mathlib arithmetic functions, beginning with the Möbius function μ and extending to the sum-of-divisors function. The sigma abbreviation is defined as sigma (k : ℕ) := ArithmeticFunction.sigma k, which computes ∑_{d|n} d^k. The local setting keeps statements minimal to support later Dirichlet inversion once interfaces stabilize.
proof idea
The proof applies native_decide directly to the expression sigma 3 5, which enumerates the divisors of 5 and sums their cubes.
why it matters
This supplies a basic numerical anchor for the sigma function inside the arithmetic-functions module. The module positions these wrappers as footholds for Möbius tools, though the declaration carries no downstream uses and does not touch Recognition Science landmarks such as the J-uniqueness or phi-ladder.
Switch to Lean above to see the machine-checked source, dependencies, and usage graph.