pythagorean_prime_onehundredthirteen
plain-language theorem explainer
113 is established as a prime congruent to 1 modulo 4, marking it as a Pythagorean prime. Researchers examining prime properties tied to framework constants would reference this verification. The proof applies a computational decision procedure to confirm both conditions simultaneously.
Claim. $113$ is prime and $113$ ≡ 1 (mod 4).
background
The module develops arithmetic functions centered on the Möbius function μ as a starting point for Dirichlet inversion techniques. The Prime predicate is the repository-local alias for the standard natural-number primality predicate from Mathlib. Upstream dependencies include foundational classes from OptionAEmpiricalProgram and structures from SimplicialLedger and GameTheory, though this theorem directly uses the Prime abbreviation from the Basic primes submodule.
proof idea
The proof consists of a single native_decide application that evaluates the primality of 113 and its residue modulo 4 through built-in decision procedures.
why it matters
This theorem supplies a concrete Pythagorean prime instance within the arithmetic functions module. It may underpin calculations involving primes of the form 4k+1, which appear in contexts like the fine-structure constant approximations near 137. No downstream usages are recorded yet, leaving its precise integration open.
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