During my interview with Arthur Benjamin, he spoke briefly about the joy of discovering something in mathematics, even if one is relatively certain that the given relationship has been known for years, or even centuries. Along the same lines, I’ve been playing with right triangles of late.
They are called Pythagorean triples—three numbers, a, b, and c, where aSquared+bSquared=cSquared, and a b and c are all whole numbers. I wanted a way to generate them.
I started by noticing that the difference between two squares, such that one root is one less than the other, is always an odd number.
2*2=4
1*1=1
4-1=3
3*3=9
2*2=4
9-4=5
4*4=16
3*3=9
16-9=7
The result climbs up the odd number line, 3, 5, 7, 9, 11…
Another way to put this, is to say that the difference between consecutive whole number squares is on the line defined by 2x+1.
This can be shown algebraically:
(x+1)Squared-xSquared=2x+1
Expanding (x+1)Squared gives us (xSquared+2x+1)
So (xSquared+2x+1)-xSquared=2x+1
Add xSquared to each side and we get:
XSquared+2x+1=xSquared+2x+1
Take xSquared from each side and we get:
2x+1=2x+1
Take 2x from each side and we get
1=1
That’s about as true as numbers can get.
I’m not the first to have noticed that relation, but I did run across it independently.
Next, in order to confine the results to whole numbers, we substitute some whole number that is greater than or equal to 1 for x, which will give you side a.
2*2+1=3
Next we square a: 3*3=9
Subtract 1: 9-1=8
And divide by 2: 8/2=4
That gives us side b=4.
Next since we know the difference in length from b to c is 1, c=b+1: 4+1=5
3, 4, 5
3*3=9
4*4=16
9+16=25
5*5=25
Let’s try using 2 in place of x. 2x+1
2*2=4, 4+1=5, a=5
5*5=25, 25-1=24, 24/2=12, b=12
12+1=13 c=13
5, 12, 13
5*5=25
12*12=144
25+144=169
13*13=169
This is much easier to do in your head than it is to write it out.
3*2=6+1=7
7*7=49-1=48/2=24
24+1=25
7, 24, 25
4*2=8+1=9
9*9=81-1=80/2=40
40+1=41
9, 40, 41
When I looked it up, it turned out that Euclid used this trick to prove that the set of Pythagorean triples is infinite, since the set of odd numbers is infinite and you can use any odd number to generate sides a b and c.
Q.E.D.
Next, I got to wondering if you could generalize the method, so that you could generate triples when the difference between side b and c, c-b could equal any whole number.
Yes you can! If d=c-b than the line that your numbers need to sit on is (2*d)x+dSquared, where x is any whole number greater or equal to 0, unless d=1.
If d=1, the line is 2x+1. If x=0, you end up with 1, 0, 1. And while 1*1+0*0 does equal 1*1, that’s not really a triangle; it’s a line segment.
By the by, when you use d=1 you get 1*2=2 and 1*1=1, so the line is 2x+1, which we already knew.
Let’s try d=2.
2*2=4 and 2*2=4, so our line is given by 4x+4
Put 0 in place of x.
4*0=0+4=4, so a=4
4*4=16-4=12/4=3 so b=3
3+2=5 so c=5
4, 3, 5, which is our old friend 3, 4, 5
Using 4x+4 and replacing x with 1
4*1=4+4=8, a=8
8*8=64-4=60/4=15, so b=15
15+2=17
8, 15, 17
Using x=2, 2*4=8+4=12, a=12
12*12=144-4=140/4=35, b=35
35+2=37
12, 35, 37
Now let’s set d to 3
3*2=6 and 3*3=9, so the line is 6x+9
Starting with 0
6*0=0+9=9, a=9
9*9=81-9=72/6=12, b=12
12+3=15
9, 12, 15
X=1
6*1=6+9=15 a=15
15*15=225-9=216/6=36, b=36
36+3=39
15, 36, 39
X=2
6*2=12+9=21, a=21
21*21=441-9=432/6=72, b=72
72+3=75
21, 72, 75
In fact, there are simpler methods for making triples, but I found this all on my own, and I can’t even describe the joy of it, when the numbers click together and it all makes sense.
As any whole number greater than 0 can be substituted for d, and the set of whole numbers greater than 0 is infinite, this adds another layer of infinity on top of the one Euclid found, just in case one infinity wasn’t enough
Q.E.D.