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Introduction
English mathematician and scientist Thomas Harriot (1560–1621) gave the usual formula for Pythagorean triples using his new algebraic notation but he also started to list them in a systematic way. If he could have continued his list indefinitely, would he have listed all of the Pythagorean triples? In exploring this question, students can recognize and describe patterns, write and use algebraic formulas, and construct proofs, including proofs by mathematical induction. Students also have the opportunity to study an historical approach in which a mathematician seemed to believe that tabular presentation of a result was just as valuable, effective, and interesting as symbolic presentation of that result.
This material could be used in an undergraduate number theory course, in a “proofs” or “transition” course, or as enrichment for bright algebra or general education students. At least part of it could be used in college algebra or other general education courses. In lower level courses, the material should be presented in class, either during an interactive lecture or as an in-class exploration. In more advanced courses, it could be presented in class, as homework, or as a project for a small group (or small groups) of students.
Mathematical Background
Pythagorean triples are ordered triples (x, y, z) of positive integers with the property that x2 + y2 = z2. Such a triple can be viewed geometrically as giving the lengths of the sides of a right triangle with hypotenuse of length z and legs of lengths x and y. The Pythagorean triple (3, 4, 5) has been known since antiquity, with the triple (5, 12, 13) making its appearance fairly early on in history as well.
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