Pythagorean triples as integer points on a circular cone

This paper investigates Pythagorean triples (a, b, c), which are solutions to the equation a² + b² = c² in natural numbers, from the perspective of their geometric interpretation as integer points on the surface of the circular cone x² + y² - z² = 0. The focus is on two key aspects: 1. Counting triples with constraint z ≤ v via the function F(v). Based on classical results in number theory, an exact formula is proven expressing F(v) through the difference between the number of divisors of the form 4k+1 and 4k+3 of the number τ². This result links the problem to the representation of numbers as sums of two squares. 2. Generation of new Pythagorean triples using parametric transformations. A method is proposed for constructing new triples (A, B, C) from an initial triple (a, b, c) by projecting lines through a point on the cone under the condition m² + n² - p² = 1. An example is given: (3, 4, 5) → (28, 45, 53). The study combines methods from analytic number theory, algebraic geometry, and algorithmic approaches, offering new perspectives for the investigation of Diophantine equations.