Pythagorean Triples?

Question

I hate to ask questions since there aren't any points in it for me.

However, out of curiosity...

What is the largest irreducible pythagorean triple you know of and is there a formula for predicting them?

By irreducibe, I mean that they can't be simplified. I.e. 6,8,10 is a pythagorean triple but it is identical to 3,4,5.

I am more interested in the formula for predicting them.

Answer 1

try this site:

http://www.math.utah.edu/~alfeld/ptt.html

Answer 2

It can be proven that there are an infinite number of Pythagorean triples. In fact, it can be proven that there are an infinite number of Pythagorean triples where the two smaller numbers differ by 1. (3,4,5 is the first, then 20,21,29, etc.)

Also, for a cute way to generate the Pythagorean triples where the two smaller numbers differ by one:

Take the sequence 1,2,5,12,29,70,169,etc., where to get the next number, you double the last number and add the number just before it.

Now pick any two adjacent numbers and plug them into the Pythagorean triple generator:

x^2 - y^2, 2xy, x^2 + y^2

For example, if 12 and 29 are selected, the Pythagorean triple calculated is (29^2 - 12^2, 2*29*12, 29^2+12^2)= (697, 696, 985).

Answer 3

If a and b are any integers, then x=a^2-b^2, y=2ab, and z=a^2 +b^2
give a pythagorean triple. If you pick a and b to be different odd primes, then (x/2,y/2,z/2) will be primitive. For example, if a=37 and b=17, then you get the triple (540, 629, 829). Clearly, you can get triples as larger as you want this way.

Answer 4

All "primitive" pythagorean triples are of the form,
[ab, (a^2 − b^2)/2, (a^2 + b^2)/2]
where a and b are relatively prime odd numbers, a > b >= 1.

And the set of all pythagorean triples are just multiples of the "primitive" ones.

Answer 5

the formulais
2m,m^2-1 and m^2+1

Answer 6

In this context, what do you mean by irreducible?