Pythagorean triangles.

[Guv]

Lior posted a list of Pythagorean triangles to a different thread, and mentioned that there are infinitely many such triangles. Since many of the ones on his list were similar to others, I am sure that he did not use the following cute algorithm.

• Choose an odd and an even integer (u, v) with no common factor and u > v.
  For example: (3, 2) (24, 5) (49, 36), but not (7, 3) or (33, 6).
  
• z = u^2 + v^2
  
• y = u^2 - v^2
  
• x = 2*u*v

x^2 + y^2 = z^2, and x, y, z will have no common factor.

[Destined Soul]

It's been a little while since I took some Number Theory courses, but I believe the word you're looking for is Pythagorean Triples, not triangles.

As well, you can better define your "algorithm" by stating that u and v must be relatively prime.

Number theory gets real fun once you've left normal real number and restrict yourself to integers, imaginary numbers, Gaussian integers, etc.

Destined

Edit: Btw, the combination (7,3) does work, as they're relatively prime.

[Guv]

Destined Soul: All pairs of numbers work, but the following rules apply.

• If the numbers are equal you get a degenerate triangle with the length of one side equal to zero.
  
• If either number is not an integer, the sides of the Pythagorean triangle will not be integers.
  
• If both numbers are odd, the resulting triple is not relatively prime.
  
• If the numbers are not relatively prime, neither is the resulting triple.

Is having no common factor wrong in the above context, or is it a synonym for relatively prime?

Does relatively prime apply to more than two integers?

[Guv]

Destined Soul: What is a Gaussian integer?

BTW: I thought that Number Theory dealt solely with integers and/or integer solutions to various equations.

[Starman]

We used to find right triangles by taking any odd integer a (>1), squaring it, halving the square, subtract 0.5 for b and add 0.5 for c.

3, 3^2, 9/2, 4.5-0.5=4, 4.5+0.5=5, giving 3,4,5

5, 5^2, 25/2, 12.5-0.5 = 12, 12.5+0.5 = 13, giving 5,12,13


Surely (7,3) are not valid in your formula as u,v are not one odd, one even - or have I missed something here?

[Starman]

My formula also works with even integers but you don't get an integer result for b and c.

[Destined Soul]

Starman: Read the definition of the 'algorithm' below. That may help.

Guv:

The formal definition is "the integers a and b are relatively prime if a and b have their greatest common divisor equal to 1."

And, yes, relatively prime does apply to more than one number - at least in principle. I know you can apply the GCD for more than one number, so it shouldn't be too hard to further that to multiple integers.

As for the numbers being equal, you cannot have them being relatively prime, as their GCD equals themselves. I think the integer 1 is a special case, usually ignored, but if you take any number > 1 then the GCD > 1, hence not relatively prime.

For them being integers, it is included in the definition of relatively prime.

Here's that little 'algorithm' you mention, quoted from my text:

"Theorem 13.1 - The positive integers x,y,z forma primitive Pythagorean triple, with y even, if and only if there are relatively prime positive integers m and n, m > n, with m odd and n even or m even and n odd, such that x = m^2-n^2, y = 2mn, z = m^2+n^2." (Elementary Number Theory and its applications, Kenneth H Rosen, 4th edition)

By a 'primitive' pythagorean triple, the mean where the GCD(x,y,z)=1. (ie relatively prime, in a sense..)

Sorry about above, though - I didn't realize that you stated one odd and one even... having them this way does guarantee a GCD = 1, but you can still find any pythagorean triple by just ensuring that u & v (or m & n) are relatively prime. Odd/Even just ensures primitiveness.

As for Number theory involving only integers, no, not really. It does, in a sick, sort of way, but is definitely not limited to. I say this, as I can represent the number 0.636363636... as 7/11 using two integers, or even in another form [0;1,1,1,3], a 'finite continued fraction.' I highly recommend picking up a book on the subject, or taking a course on it if you can, so long as this sort of thing interests you.

Numbers get beyond normal integers, and even rationals. A Gaussian Integer is an integer Z, where Z = a + bi, where a,b are normal integers (0, +/-1, +/-2, etc.) and i = square root of (-1). Defining prime numbers in this is quite fun as well.

I'm sure I could find another formula or two to generate pythagorean triples, but it's my bedtime. :P

Destined

[Guv]

Starman: Never saw your algorithm. Note that it only generates a subset of the possible Pythagorean triples. It looks as though the triplets generated by your algorithm always have the largest number one greater than the even number in the triplet.

The (u, v) algorithm can generate them all if you have the time. With a little ingenuity, you can generate triples which satisfy special conditions.

Destined Soul: Thanx for the information. Number Theory is interesting to me, but not interesting enough to spend time increasing my meager knowledge of it.