Which triples of whole numbers {a, b, c} satisfy
a^{2} + b^{2} = c^{2} ?
Such triples are called Pythagorean triples
because they are integer solutions to the Pythagorean theorem.
You probably know {3, 4, 5} and {5, 12, 13}. But can
you classify all possible Pythagorean triples?
Answer: it is possible to prove that all
Pythagorean triples are of the form
{ M^{2}N^{2}, 2MN, M^{2}+N^{2} }
for some integers M and N, or they are
multiples of this form.
Thus setting M=2,N=1 gives {3,4,5} and M=3,N=2
gives {5,12,13}.
Presentation Suggestions:
If you are really motivated and have time to practice this,
you can try to following.
Before telling students the rule for construction,
tell them to give you
any number and that in your head
you will construct a Pythagorean triple using
that number. If they give you an even number K=2M,
let N=1; if they give you an
odd number K=2N+1, let M=N+1. If you can do this quickly
for several examples, you can say "Well, since I'm
not that good with mental calculations, there's obviously
a trick. It turns out that all Pythagorean triples are
of this form..."
The Math Behind the Fact:
Simple number theory arguments using parity will give
this conclusion. Assume
a^{2} + b^{2} = c^{2}
for an integer triple (a, b, c). By removing any common factors, if needed,
we may assume a, b, and c have no common factor.
Since odd perfect squares must be congruent to 1 mod 4, and even squares are congruent to 0 mod 4, we can conclude that c must be odd, and at exactly one of a or b must be even. Suppose b is even. Then b=2k for some integer k, hence
4k^{2} = b^{2} = c^{2}  a^{2} = (c+a)(ca).
Since (c+a) and (ca) must have the same parity (evenness or oddness), they must both be even.
Then c+a=2r, ca=2s and rs=k^{2}. It is easy to check that c=r+s, and a=rs. But r and s can have no common factors because otherwise c and a would both share that common factor as well. So they must both be perfect squares, say a=M^{2} and b=N^{2}. This gives the desired result.
How to Cite this Page:
Su, Francis E., et al. "Pythagorean Triples."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.

