isolated_prime_onehundredseventythree
plain-language theorem explainer
173 is prime while 171 and 175 are composite. Number theorists checking small isolated primes or prime gaps would cite this explicit instance. The proof is a one-line computational decision that directly evaluates the primality predicates.
Claim. $173$ is prime while $171=3^2×19$ and $175=5^2×7$ are composite, i.e., $Prime(173)∧¬Prime(171)∧¬Prime(175)$ where $Prime(n)$ asserts that $n$ is a prime number.
background
The declaration sits inside the module on arithmetic functions that supplies lightweight wrappers around Mathlib's Möbius function and related Dirichlet tools. Upstream results define the set of primes as the subset of natural numbers satisfying the standard primality predicate and introduce a transparent alias for that predicate. The local setting keeps statements minimal so that deeper inversion formulas can be added once the basic interfaces stabilize.
proof idea
The proof is a one-line wrapper that applies native_decide to evaluate the three primality claims by direct computation on the concrete integers.
why it matters
The result supplies a concrete verified instance of an isolated prime that can be referenced in larger arithmetic constructions inside the Recognition Science framework. It fills a specific slot in the primes module even though no downstream theorems currently invoke it. The check aligns with the need for explicit small-integer facts when building toward mass ladders or arithmetic-function identities.
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