pythagorean_prime_five
plain-language theorem explainer
Five is both prime and congruent to one modulo four, confirming it as a Pythagorean prime. Number theorists examining sums of two squares or quadratic reciprocity reference this base case. The verification applies a direct computational decision procedure that evaluates the predicates without case analysis.
Claim. $5$ is prime and $5$ satisfies $5 ≡ 1 mod 4$.
background
The module supplies lightweight wrappers around Mathlib arithmetic functions, beginning with the Möbius function, while keeping statements minimal until Dirichlet inversion layers are added. Prime is imported as a transparent abbreviation for the standard natural-number primality predicate. The local setting therefore treats basic prime facts as scaffolding before deeper arithmetic-function identities are developed.
proof idea
The proof is a one-line wrapper that invokes the native_decide tactic to evaluate both conjuncts by direct computation.
why it matters
This supplies a concrete Pythagorean-prime instance inside the number-theory foundations of the module. It supports later work on sums of squares, though the dependency graph lists no immediate parent theorems. Within Recognition Science it adds arithmetic scaffolding without touching the forcing chain, phi-ladder, or RCL.
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