9860

Pythagorean Primitive Triples Using Primes

Primitive Pythagorean triangles (PPTs) are right triangles that have integer sides with no common divisor. They were used by the Babylonians for use in excavation and surveying. Methods for finding the triples that represent PPTs were valued. A novel method is shown here.
Any (and all) Pythagorean triples can be derived starting from two integers and . You can change the values of and by multiplying their current values by prime numbers. To do this, first click one of the two reset buttons and then repeat the following two steps:
1) choose a prime value from the setter bar at the top.
2) click one of the buttons labeled "A twice", "B twice", or "both A and B".
To generate PPTs, keep and relatively prime as described in the details.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

The early Greeks used Euclid's formula to calculate PPTs from two relatively prime integers and , where : .
The parametrization offered here is similar and amounts to a change of variables. Let and be relatively prime and . Then is a PPT.
Not only is this parametrization very pretty, it can be generalized and has been used in the examination of Fermat's last theorem.
In order to generate only PPTs, and must be relatively prime.
1) Never press the "both A and B" button.
2) Never include a prime as an additional factor in or if that prime is already a factor of the other integer.
Select the checkbox "enforce rules:" to be warned that the relatively prime rule is violated.
The prime factors of the resulting PPT sides can be examined by mousing over the number of interest.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+