coprime_eight_thirtyseven
plain-language theorem explainer
Eight and thirty-seven share no common prime factors. Number theorists applying arithmetic functions within Recognition Science cite this instance when handling moduli that mix powers of two with the prime 37. The proof reduces the claim to a direct computational check of the gcd via the native_decide tactic.
Claim. The positive integers 8 and 37 are coprime: $8$ and $37$ share no common prime factors, or equivalently $gcd(8,37)=1$.
background
The module supplies lightweight wrappers around Mathlib arithmetic functions, beginning with the Möbius function μ. Statements remain basic while deeper Dirichlet algebra is prepared. The declaration depends on the upstream theorem and, which supplies an explicit log-derivative bound M that produces the angular Lipschitz constant logDerivBound = M * r on a circle of radius r.
proof idea
The proof is a one-line wrapper that invokes the native_decide tactic to evaluate the coprimality predicate by direct computation of gcd(8,37).
why it matters
This supplies a concrete coprimality fact inside the arithmetic functions module. It aligns with the eight-tick octave landmark (period 2^3) because 8 = 2^3 and may support later Möbius or squarefree calculations among the sibling definitions. No downstream theorems are recorded yet.
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