jonas2 1228239293|%e %b %Y, %H:%M %Z|agohover
Taken from the Wolfram website. I am pretty sure this method wasn't covered but I can't always be sure…
If 
is a Primitive Pythagorean Triple, then 
generates a new primitive triple 
where
.
Let's exam a simple example. We were told that 
is a primitive triple. So…
…which can easily be verified to be valid!
Greg Gifford 1228452111|%e %b %Y, %H:%M %Z|agohover
There's also another way to generate primitive triples that I don't think was discussed in class, although it isn't linear algebra. If you let 
(any odd number), then 
and 
. To prove it:
Greg Gifford, 1228452111|%e %b %Y, %H:%M %Z|agohover
eguzman2 1228632926|%e %b %Y, %H:%M %Z|agohover
Here is something that I wanted to tell you about in class but didn't not had time:
Plato’s Formula:
Let m be contained the set of integers with m > 1. Then we can generate a Pythagorean triple by letting a = 2m, b = m^2 – 1, c = m^2 + 1.
Ex. Let m = 2, so we have that a = 2(2) = 4, b = 4 – 1 = 3, c = 4 + 1 = 5.
Then 4^2 + 3^2 = 5^2, which is true.
The formula generates finitely many triple but not all Pythagorean triples.
Formula for generating all Pythagorean triples
Consider the Pythagorean triple x – y – z where (x,y,z) = d. Let x = du,
y = dv, and z = dw where (u,v,w) = 1. So we have u^2 + v^2 = w^2, thus
u – v – w is a triple. Then we can say that every Pythagorean triple is a multiple of primitive triple.
Ex. Let x = 6, y = 8, z = 10. We have that 6^2 + 8^2 = 10^2. Then (6,8,10) = 2. We have that:
x = 2(3) = 6
y = 2(4) = 8
z = 2(5) = 10
This tells that 3 – 4 – 5 is a Pythagorean triple, which we know is true.
Using this idea in the equations for generating infinitely many primitive triples that Dan showed we can generate all triples by adding a parameter d. So we have:
a = d*(m^2 – n^2), b = d*(2mn), c = d*(m^2 + n^2) where m, n, d are contained in the set of integers with m>n>0 and d positive, n or m is odd and (m,n) = 1.
Conclusion:
The formula generates all Pythagorean triples but not uniquely.
eguzman2, 1228632926|%e %b %Y, %H:%M %Z|agohover
