Formulas For Generating Pythagorean Triples - Progressions of Whole and Fractional Numbers

Progressions of Whole and Fractional Numbers

The German monk and mathematician Michael Stifel published the following method in 1544.

Consider the progression of whole and fractional numbers:

The properties of this progression are: (a) the whole numbers are those of the common series and have unity as their common difference; (b) the numerators of the fractions, annexed to the whole numbers, are also the natural numbers; (c) the denominators of the fractions are the odd numbers, etc.

To calculate a Pythagorean triple select any term of this progression and reduce it to an improper fraction. For example, take the term . The improper fraction is . The numbers 7 and 24 are the sides, a and b, of a right triangle, and the hypotenuse is one greater than the largest side. For example:

Jacques Ozanam republished Stifel’s sequence in 1694 and added the similar sequence with terms derived from . As before, to produce a triple from this sequence, select any term and reduce it to an improper fraction. The numerator and denominator are the sides, a and b, of a right triangle. In this case, the hypotenuse of the triple(s) produced is 2 greater than the larger side. For example:

Together, the Stifel and Ozanam sequences produce all primitive triples of the Plato and Pythagoras families respectively. The Fermat family must be found by other means.