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Pythagorean Triples

A "Pythagorean Triple" is a set of three numbers a,b,c such that a²+b²=c².  Pythagoras proved that the sides of a right triangle have this relationship, then went crazy after discovering that the sides of a right triangle aren't always rational.

At first sight, it appears that this is a fairly dull topic -- there are lots of integer solutions to the Diophantine (named after Diophantus, 3rd century A.D. Greek mathematician of Alexandria) equation a²+b²=c², such as 3,4,5; and 5,12,13.  Big deal.

Oh, look -- one of the numbers is a multiple of 4 and the other two are odd.  It turns out that's always the case when the triple has no common factor (a primitive triple).  Why is that?  Interesting...

And check this out -- not only is c² the sum of two squares, so is c.  What?!  Yes, it's true.  The average of the two odd numbers is one of the two squares.  Take 3,4,5 -- the average of the two odd numbers is 4, which is one of the two squares that adds up to 5.  Half of the difference of the two odd numbers is the other square to form the sum, 5.  Come on -- admit it.  There's more to Pythagorean Triples than you ever expected!

In this section, we will explore some of the interesting facts relating to Pythagorean Triples.  One of the most interesting areas is the "Infinite Descent" which is a style of proof in which we start with the smallest number with a certain property, and then prove that there is a smaller number with that property -- therefore there is really no number with that property.  The fact that the big number in a Pythagorean Triple is itself a sum of two squares is often a key to such proofs.

Contents of this section:

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Internet references

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Related Pages in this website

The Pythagorean Theorem, in the Geometry section

Infinite Descent Proofs

Number Theory


Formulas for Primitive Pythagorean Triples and their deriviation -- a way to generate all the triples such that a^2 + b^2 = c^2

Prove that the area of a right triangle with integer sides is not a perfect square.  The proof is here (with some help from someone from the nrich website)

Puzzle question -- if m is the product of n distinct primes, how many different right triangles with integer sides have a leg of length m?

Arithmetic Sequence of Perfect Squares, page 3 -- If a^2, b^2, c^2 are in arithmetic sequence, why is their constant difference a multiple of 24?  Look at the second answer to this question for the relationship between Pythagorean Triples and this arithmetic sequence of squares.

Theorems Involving Perfect Squares -- answers questions such as "why is the square root of x irrational unless x is a perfect square?" and other fundamental questions about perfect squares.

The webmaster and author of this Math Help site is Graeme McRae.