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“Brahmagupta's bhāvanā lemma (Brahmasphutasiddhanta XVIII.64-65, 628 CE; rendered §76-77 in Colebrooke 1817): if (x₁, y₁) solves x²N + k₁ = y² and (x₂, y₂) solves x²N + k₂ = y², then their cross-product (x₁y₂ ± x₂y₁, y₁y₂ ± Nx₁x₂) solves x²N + (k₁k₂) = y². The mathematical foundation Bhāskara II's chakravala (1150 CE) stands on. Lagrange's 1768 European solution to "Pell's equation" rediscovers the same composition + iteration structure. ”