mod4_thirtyone_prime
plain-language theorem explainer
31 is a prime congruent to 3 modulo 4. Number theorists working with arithmetic functions or Dirichlet characters reference this concrete instance when checking small cases in residue classes. The proof reduces to a single native decision step that evaluates both primality and the residue directly.
Claim. $31$ is prime and $31 ≡ 3 mod 4$.
background
The module supplies lightweight wrappers around Mathlib arithmetic functions, beginning with the Möbius function. Prime is the local transparent alias for the standard predicate Nat.Prime on natural numbers. This theorem records a specific prime in the 3 mod 4 class that can appear in square-free checks or inversion formulas. The upstream result is the Prime abbreviation, which imports Mathlib primality without extra hypotheses.
proof idea
The proof is a one-line wrapper that applies the native_decide tactic to evaluate the conjunction by direct computation.
why it matters
This supplies a verified small prime fact for arithmetic function identities in the primes submodule. No parent theorems currently depend on it, yet it supports the module goal of stabilizing basic interfaces before layering Dirichlet algebra. It provides a concrete number-theoretic primitive consistent with the Recognition framework's use of small cases in forcing chains and composition laws.
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