Contents Supplements and Links
Objectives: To systematically develop an understanding of integral solutions to the equation x2 + y2 = z2, which serves as a (tractable) example of pursuits in algebraic number theory while touching on many elementary concepts of the field. Necessary Background: No background is necessary, but a certain level of mathematical maturity is required to undertake these investigations. Section III is more difficult than Section IV, but it is more central to the questions that number theory tends to ask, so it has been given preference in the ordering.
Summary: In Section I, students generate (or examine) lists of Pythagorean triples and reduce them to simpler cases. Section II explores some of the properties of primitive triples. In Section III, students count the number of triples of a given hypotenuse. In Section IV, students find all triples with certain restrictions. Section V suggests an analogous project for Gaussian integers.
This project, more than the others, is open-ended. The questions are intended as guides, but if they become cumbersome and confusing, feel free to ignore them and pursue the project in your own direction.


I. Introduction

The Pythagorean theorem is named for the Greek mathematician Pythagoras, who lived in the 6th century BCE, though the theorem had been known elsewhere for some time before.

Theorem 1 (Pythagorean Theorem and converse)  Let x, y, and z be positive numbers. Then z is the length of the hypotenuse of a right triangle with side lengths x, y, and z if and only if

(1)

We will be interested in solutions to (1) in which x, y, and z are all integers. A Pythagorean triple is an ordered triple (x, y, z) of three positive integers such that x2 + y2 = z2. If x, y, and z have no common divisors, then the triple is called primitive.

For example, (3, 4, 5) is a primitive Pythagorean triple, as 32 + 42 = 52 and the sides are relatively prime. Find some other Pythagorean triples.


Using a programmable calculator or computer program, generate a systematic list of solutions to x2 + y2 = z2. (Programs in a variety of formats are available here, and here is a list of the first hundred primitive Pythagorean triples.)


What do you notice about the solutions? Can you optimize the program in any way? Can some solutions be reduced to others?


In the reduced set of solutions, what do you notice about the values of z? Might we make similar statements about x and y? Can you prove that these observations always hold?


What kinds of numbers take on the values of z – x, z – y, and |x – y| for all triples?


Use these observations to create "generators" for Pythagorean triples, i.e. formulas for x, y, and z in terms of the generators r and s that will give a Pythagorean triple for any integers r and s.


II. Properties of r, s, x, y, and z


What conditions on r and s guarantee that the triple they generate is primitive? Does the converse hold — that the generators of every primitive triple satisfy these conditions? Can you prove these statements?


How many primitive Pythagorean triples exist?


If (x, y, z) is a primitive Pythagorean triple, can we make any statements about at least one of the sides being divisible by 3, 4, or 5? Is z – 1 consistently divisible by any number? Can you prove these observations?


III. Counting Triples


The only primitive Pythagorean triples with hypotenuse 5 have legs 3 and 4. Are there any two primitive Pythagorean triples that have the same hypotenuse but different legs? Find several examples, and make a conjecture about when this happens.


Try to determine experimentally whether there is there any natural number z that is the hypotenuse of exactly three primitive Pythagorean triples. What about exactly four triples? exactly five?


Make a conjecture about the number of primitive Pythagorean triples with a given hypotenuse z. Use this to find numbers which are the hypotenuses of many (at least five) primitive Pythagorean triples.


Investigate the following: How many primitive Pythagorean triples have a given natural number as a leg?


How many primitive Pythagorean triples have a given natural number as a side (either as a leg or as the hypotenuse)?


How many (not necessarily primitive) Pythagorean triples have a given number as a hypotenuse?


IV. Restrictions on Primitive Triples


Which triples satisfy z – y = 1? Find a formula for the nth such triple.


More generally, which primitive triples satisfy z – y = b for a fixed odd square b? Which satisfy z – x = a for a fixed number a that is twice a square?


What is the nth triple that satisfies |x – y| = 1?


What is the nth triple that satisfies |x – y| = 7?


In general, what values d are possible for the difference |x – y|?


Find all triples with a given d = |x – y|.


V. Gaussian Triples


So far we have been looking at Pythagorean triples with integer solutions. The Gaussian integers are an extension of the integers into the complex numbers. A Gaussian integer is a complex number a + bi, where a and b are ordinary integers and i2 = –1. What solutions in Gaussian integers can you find to the equation x2 + y2 = z2? All of the integer solutions are all also Gaussian solutions (with the imaginary parts equal to 0), but there are many more! For example, (1 + 4i)2 + (8 – 4i)2 = (7 – 4i)2. How much of the above analysis carries through to this more general case? (Hint: Quite a lot!)


References

Relevant links: