Machin-Type Formulas and Metallic Pythagorean Triples: A Gaussian Classification

Machin-type arctangent formulas are completely classified inside the metallic Pythagorean family via Gaussian valuations, with explicit finite modules for small prime sets that recover the classical identities of Machin, Euler, Gauss and others. For an integer n ≥ 1 let θ_n = 2 arctan(2/n). For n ≥ 3 this is an acute angle of a primitive Pythagorean triple, and cot(θ_n/4) is the n-th metallic mean. We study integer relations ∑_i c_i θ_{n_i} ∈ (π/2) ℤ among these metallic angles. Writing θ_n = 2 arg(n + 2i), we show that such a relation holds exactly when a signed Gaussian valuation vanishes at every odd prime, and that its constant is controlled modulo 2 by the triples with c − b = 1; in particular every relation involving only triples with c − b ∈ {2, 8} sums to a multiple of π. For a finite set S of primes, the parameters n for which n² + 4 has all its odd prime factors in S form a finite set N_S, which we determine effectively by reducing to Pell equations and applying Carmichael’s primitive divisor theorem. We compute N_S for S = {5}, {13}, {5, 13} and {5, 13, 17}, and give explicit ℤ-bases of the relation modules for the first three. The two-term formulas of Euler, Hermann, Hutton and Machin, and the three-term formulas of Strassnitzky and Gauss, all lie in the module for S = {5, 13}; for example, Machin’s formula reads 4∠(5, 12, 13) − ∠(239, 28560, 28561) = π/2. The method is classical; the contribution is the metallic-triple interpretation, the inclusion of odd parameters, the parity law, and complete determinations for small S.