Mudd Math Fun Facts!
hosted by the Harvey Mudd College Math Department created, authored and ©1999-2007 by Francis Su
You can now subscribe to our RSS feed to get the latest Fun Facts!
Get a random Fun Fact!
No subject limitations
Search only in selected subjects
    Algebra
    Calculus or Analysis
    Combinatorics
    Geometry
    Number Theory
    Probability
    Topology
    Other subjects
  Select Difficulty  
Enter keywords 

  List All : List Recent : List Popular
  About Math Fun Facts / How to Use
  Contributors / Fun Facts Home
© 1999-2007 by Francis Edward Su
All rights reserved.

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>.

Bookmark this page on: | Digg this! | Del.icio.us | Technorati | Reddit | Fark | Squidoo | Furl | Blinklist | Yahoo MyWeb | Google | Stumbleupon |

Keywords:    Pythagoras
Subjects:    number theory
Level:    Medium
Fun Fact suggested by:   Francis Su
Suggestions? Use this form.
3.95
 
current
rating 		Click to rate this Fun Fact...
    *   Awesome! I totally dig it!
    *   Fun enough to tell a friend!
    *   Mildly interesting
    *   Not really noteworthy
and see the most popular Facts!

Want another Math Fun Fact?

For more fun, tour the Mathematics Department at Harvey Mudd College!