Pythagorean Triples


Right Triangle

A Pythagorean triple is a triplet (a,b,c), where a, b, c are integers such that a^2+b^2=c^2.

It is called a Pythagorean triple because it can represent 3 sides of a right triangle. You probably already know two Pythagorean triples: (3,4,5) and (5,12,13). Are there other triples? How do we find them?

Let’s consider the equation
where a, b, c are integers. If a, b, c have a common factor d, say a=da_1, b=db_1, and c=dc_1. Then we have
which is true if and only if
So let’s assume that a, b, c have no common factors. Such triples will be called primitive. Since odd perfect squares are congruent to 1 mod 4 and even squares are congruent to 0 mod 4, then c must be odd and exactly one of a or b is even. Let’s suppose that b is even: b=2k for some integer k. Then
So both c + a and c - a must be even. Say c + a = 2r and c - a = 2s. Then rs=k^2 and we have c = r + s and a = r - s. Since a and c have no common factors, we must have that r and s have no common factors. Since rs is a square, then both r and s are squares. Say r=m^2 and s=n^2. So a=m^2-n^2, b=2mn, c=m^2+n^2 if (a,b,c) is a primitive triple.

Hence, any Pythagorean triple, with b even, is either of the form (m^2-n^2,2mn,m^2+n^2) or it is a multiple of that form. You can plug in different values for m and n to get different right triangles with integer sides. Fun!


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