Formulas For Generating Pythagorean Triples - Generating Triples Using Quadratic Equations

Generating Triples Using Quadratic Equations

There are several methods for defining quadratic equations for calculating each leg of a Pythagorean triple. A simple method is to modify the standard Euclid equation by adding a variable x to each m and n pair. The m, n pair is treated as a constant while the value of x is varied to produce a “family” of triples based on the selected triple. An arbitrary coefficient can be placed in front of the “x” value on either m or n, which causes the resulting equation to systematically “skip” through the triples. For example, let’s use the triple which can be calculated from the Euclid equations with a value of m = 5 and n = 2. Also, let’s arbitrarily put the coefficient of 4 in front of the “x” in the “m” term.

Let and let

Hence, substituting the values of m and n:

Note that the original triple comprises the constant term in each of the respective quadratic equations. Below is a sample output from these equations. Note that the effect of these equations is to cause the “m” value in the Euclid equations to increment in steps of 4, while the “n” value increments by 1.

x side a side b side c m n 0 20 21 29 5 2 1 54 72 90 9 3 2 104 153 185 13 4 3 170 264 314 17 5 4 252 405 477 21 6