# Math Research Project — Pythagorean Triples

Each Math Research Project outline consists of a Seed Question, Questions to Build On, Extensions, and Basic Background.  Click here to see a list of available Math Research Project outlines.

The Seed Question is a question that appears in a typical math curriculum.  Questions to Build On are simple extensions, re-interpretations, and generalizations of the Seed Question that a student can build a simple math research paper around, with help from a mentor (be it student or teacher).

The Extension Questions are bigger, more challenging extensions of the Seed Question.  If a student can make some headway into understanding these, great!  If not, they are good questions to include as part of the “Where do we go from here?” section of the paper.

The Basic Background consists of some of the mathematical leg-work that underlies the investigation.  It can be worked on simultaneously with the investigation, which gives it context

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### Pythagorean Triples

Seed Question

Find some triples of positive numbers that satisfy the equation $a^2 + b^2 = c^2$

Questions to Build On

Find all triples of positive numbers with c < 100 that satisfy $a^2 + b^2 = c^2$.

Show that a multiple of a Pythagorean Triple is also a Pythagorean Triple.

Find all Pythagorean Triples whose legs are consecutive integers.

Find all Pythagorean Triples whose legs differ by two.

Find all Pythagorean Triples that consist of a pair of consecutive integers.

Extension Questions

Show that a primitive Pythagorean triple always consists of at least one multple of three.

Find some primitive Pythagorean quadruples.

Find some integer solutions to the equation $a^2 + b^2 = c^3$

Basic Background

Verify a number of Pythagorean Triples.

Prove the Pythagorean Theorem (in multiple ways!).

Prove Euclid’s Formula.

Click here to see a list of available Math Research Project outlines.

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