Pell Equation. V. Systematic relation between the Pythagorean triples and Pell equations

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  • Pell equation 5 Systematic relation between the Pythagorean triples and Pell equations

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Systematic relations between the algebra of the Pell equations, x^2 - Dy^2 = 1 (called Pell-1) and x^2 - Dy^2 = -1 (called Llep-1), and the geometry of Pythagorean triangles or Pythagorean triples (PTs) are discussed. Although Llep-1 is solvable only for a limited number (though extending to infinity) of D values, such an algorithm is obtained that can construct a series of PTs corresponding to each D and involving rational number approximation of the square root of D. In the case of Pell-1, which is solvable for all square-free D, a simple algorithm is found for odd D, whereas some modification is necessary for even D. For each series of PTs thus obtained interesting properties regarding their recursive relations are found.

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