Polynomial curves on the Pythagorean quadric and their q-analogues

(2026) 6, pp. 65-68. (pdf)

Thad S. Morton

Abstract:
This paper presents a one-parameter embedding of integers into the quadratic hypersurface x² + y² + z² = w², given by (n, n+1, n(n+1), n²+n+1). This embedding defines an exact algebraic curve lying entirely within the integer solution set of the equation. The identity admits a consistent symmetric q-analogue that preserves a nonlinear recurrence structure equivalent to a hyperbolic representation. The resulting framework connects polynomial Diophantine embeddings with q-analog recurrence relations of Chebyshev type.