pythagorean_prime_fiftythree
plain-language theorem explainer
53 is established as a prime congruent to 1 modulo 4. Number theorists working on sums of two squares or quadratic residues reference this concrete case. The proof reduces to a direct computational check via the native_decide decision procedure.
Claim. The integer 53 is prime and satisfies $53 ≡ 1 (mod 4)$.
background
The module supplies lightweight wrappers around Mathlib arithmetic functions, beginning with the Möbius function μ. The Prime predicate is the repo-local alias for the standard Nat.Prime notion of primality on natural numbers. Upstream results supply the primality definition together with structural hypotheses ensuring collision-free or tautological properties in foundation modules.
proof idea
The proof is a one-line term-mode wrapper that applies the native_decide tactic to evaluate the conjunction of primality and the modular condition directly.
why it matters
This supplies a verified instance of a Pythagorean prime inside the arithmetic-functions module. It contributes concrete number-theoretic facts that can support later constructions involving primes, though it records no downstream usages. The surrounding module context keeps statements lightweight pending deeper Dirichlet algebra.
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