Pythagorean Triples – Part 1

Slideshow 38, MathematicsMr. Richard Sasaki, Room 307

ObjectivesObjectives• Understand the definition of a Pythagorean

triple• Be able to find Pythagorean triples• Learn relations between Pythagorean triples

and their rules and regulations through a project

Definition & MeaningDefinition & MeaningWhat is a Pythagorean Triple?Three positive integers that satisfy , written in the form .An example of a Pythagorean Triple is .(3 ,4 ,5)This is generally well known and refers to a “ triangle”.

345

If we learn and if one number is missing, we can easily identify it as common knowledge.

Of course, the larger number must represent the hypotenuse. 3

5?

is not a Pythagorean Triple., however would be fine.

Factors & MultiplesFactors & MultiplesIf is a Pythagorean triple, then is also a triple. But is not because .

Conversely, if is a triple, and are also triples if .

A Pythagorean triple that cannot be simplified is called a primitive Pythagorean triple.Next, we will look at how to produce triples. You do not need to learn the given proof, it is simply for you to see where the following formulae come from.Note: Parts of the proof are removed as you will discover such elements of triples in your project.

Proof (Euclid’s Formulae for Triples)Proof (Euclid’s Formulae for Triples)Consider a triple that satisfies where .

By manipulation, .Hence, .

As . for some

Also as , .

The next step is algebra manipulation! Use the worksheet to try and manipulate the expressions in purple to make and .

Proof (Euclid’s Formulae for Triples)Proof (Euclid’s Formulae for Triples)①, ②

Write ① as and ② as .

Let’s substitute ① into ② and ② into ①.② ①

𝑐−𝑏𝑚𝑛

+𝑐

𝑏=𝑛𝑚

𝑎+𝑏𝑛𝑚

+𝑎

𝑏=𝑚𝑛

2𝑐−𝑏𝑚𝑛

=𝑏𝑛𝑚2𝑎+

𝑏𝑛𝑚

=𝑏𝑚𝑛

2𝑐=𝑏𝑛𝑚

+𝑏𝑚𝑛2𝑎=

𝑏𝑚𝑛−𝑏𝑛𝑚

Proof (Euclid’s Formulae for Triples)Proof (Euclid’s Formulae for Triples)2𝑐=𝑏𝑛2

𝑚𝑛+𝑏𝑚

2

𝑚𝑛2𝑎=𝑏𝑚2

𝑚𝑛−𝑏𝑛2

𝑚𝑛2𝑐𝑏

=𝑚2+𝑛2

𝑚𝑛2𝑎𝑏

=𝑚2−𝑛2

𝑚𝑛𝑐𝑏

=𝑚2+𝑛2

2𝑚𝑛𝑎𝑏

=𝑚2−𝑛2

2𝑚𝑛Assuming the numerators and denominators are in their simplest form…𝒂=𝒎𝟐−𝒏𝟐 ,𝒃=𝟐𝒎𝒏 ,𝒄=𝒎𝟐+𝒏𝟐

These are Euclid’s formulae and can be used to calculate Pythagorean Triples.

AnswersAnswers

(3 ,4 ,5) (7 ,24 ,25)

(16 ,30 ,34 )(140 , 48 ,148)

(32 ,24 , 40) (33 ,56 ,65)

(55 , 48 ,73)(45 ,108 ,117)

(36 ,160 ,164)(75 ,308 ,317)

The results would just be negative.

Primitive Primitive

Primitive

Primitive Not Primitive

PrimitiveNot primitive Not primitive

Not primitive

Not primitive

ProjectProjectSo now you know about how to find triples!We will start a project and all complete the same project in pairs or individually.As usual, each of you will at the end need to declare who did what. You may use a calculator if you wish to and may use a computer in your own time but not in class.This is not a research project, it is for discovery. The project must be written on paper. Multiple sheets are expected.Everything will be scanned so please use paper clips, not staples.

Structure & ExpectationsStructure & ExpectationsThe project must contain the following, in order:1. Title – Must be clear & Relate to Pythagorean Triples & student names / class2. Introduction – A paragraph about what you will do and hope to do in your project3. Hypothesis – What you expect to see, any relationships or patterns between and

Structure & ExpectationsStructure & Expectations4. Testing – Calculating results, minimum should be valid results (positive triples) for . Tables may be appropriate.5. Results – Patterns that you can see, relationships, when results are primitive / not primitive, facts about primitive triples & not primitive triples, odd / even numbers6. Conclusion – How was your hypothesis? Anything you wish you did / didn’t do / changed?

Structure & ExpectationsStructure & ExpectationsScruffiness, crossings out and messy rubbings will be treated harshly. If you need to, make a first draft.A dark pencil or one of those cool pens that rub out is recommended. Be willing to add colour, avoid empty spaces and use a ruler when necessary! Use one-sided blank A4 white paper and guidelines as a guide only.This will be due on our first lesson back after the winter holiday. Only work with a friend if you are able to work together during the holiday!