Generation of Pythagorean numbers

Last Up date: 2007 July 6
Recent changes: Added link to my Near Isosceles list.
Started on 2003 November 9

This is from an email I had sent to a friend.

Hi: An exercise in arithmetic.

Chose two positive integers: m and n such that: They have no common divisor, m>n and further, one must be even when the other is odd. Examples for (m,n) would be: (2,1) (3,2) (4,1) (4,3) (5,2) (5,4) (6,1) but not (6,3) because 3 divides both, (6,5) (7,2) and so on.

Let x=m2 - n2, y=2mn, and z=m2 + n2

You will find that for all such (m,n) pairs: x2 + y2 = z2


Cheers, LaFarr
www.lafarr.com
    A couple examples using m and n:
  1. If you use m=2 and n=1 you will get: x=3 y=4 and z=5 The clasic 3, 4, 5 right triangle. Or scaled up: The 6, 8, and 10 which we all know.
  2. For m=3, n=2 we get x=5 y=12 z=13
  3. Using m=37 and n=18 we get: x=1045 y=1332 and z=1693 which are sides of a right triangle. I bet you didn't know that one!

This morning (Sun 2003-11-09) I decided to see if I could write a small program, Assembly Language of course, to generate (m,n) values. The following is the result of a short run of that program. Note: The first column is the value of m followed by all the valid values of n. The values of n that have a common divisor with m have been eliminated. These values generate Root Pythagorean numbers. They have no common divisor.

2 1
3 2 
4 1 3 
5 2 4 
6 1 5 
7 2 4 6 
8 1 3 5 7 
9 2 4 8 
10 1 3 7 9 
11 2 4 6 8 10 
12 1 5 7 11 
13 2 4 6 8 10 12 
14 1 3 5 9 11 13 
15 2 4 8 14 
16 1 3 5 7 9 11 13 15
17 2 4 6 8 10 12 14 16 
18 1 5 7 11 13 17 
19 2 4 6 8 10 12 14 16 18 
20 1 3 7 9 11 13 17 19 
21 2 4 8 10 16 20 
22 1 3 5 7 9 13 15 17 19 21 
23 2 4 6 8 10 12 14 16 18 20 22 
24 1 5 7 11 13 17 19 23 
25 2 4 6 8 12 14 16 18 22 24 
26 1 3 5 7 9 11 15 17 19 21 23 25 
27 2 4 8 10 14 16 20 22 26 
28 1 3 5 9 11 13 15 17 19 23 25 27 
29 2 4 6 8 10 12 14 16 18 20 22 24 26 28 
30 1 7 11 13 17 19 23 29 
31 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

As you can see there is a set of Pythagorean numbers, for every m>1. I am amazed that Pythagoras himself proved the existence of infinitely many such sets of Pythagorean triples. Unfortunately, I do not know his proof.

Here are lots of values for x, y, z Several very select values.

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