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From the Fun Fact files, here is a Fun Fact at the Medium level:

Pythagorean Triples

Which triples of whole numbers {a, b, c} satisfy

a2 + b2 = c2 ?

Such triples are called Pythagorean triples. 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

{ M2-N2, 2MN, M2+N2 }
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.

How to Cite this Page:
Su, Francis E., et al. "Pythagorean Triples." Mudd Math Fun Facts. <http://www.math.hmc.edu/funfacts>.

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Keywords:    Pythagoras
Subjects:    number theory
Level:    Medium
Fun Fact suggested by:   Francis Su
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