I guess it was the shirt I was wearing that reads "Math is my bag" on the
front that brought this gentleman over to introduce himself today. My wife
and I were at the gym doing our workouts at the time. Although he was
not a Mathematician, numbers fascinated this man as they do me. He said
that he had noticed that if you square an odd number and use ½ of the even
numbers on each side of that square you get the three lengths of a right
triangle. He wanted to now if that was always true. I said starting with 3
and getting greater that is true for all odd numbers.
From above 9² = 81 ½(80) = 40 ½(82) = 41
Thus 9² + 40² = 41²
He then asked me about even numbers. I said, "do you mean 8–15–17 for
example". "Yes," he said. then added. "How do we get those"?
I had to remind him of a method of always getting a Pythagorean Triple.
You chose any two integers greater than 0, for example a and b where
a > b. Now enter these numbers into the following three phrases.
1. a² + b²
2. a² – b²
3. 2ab
Example: a = 7 and b = 4
a² + b² = 49 + 16 = 65
a² – b² = 49 – 16 = 33
2ab = 2(28) = 56 ➔ 33² + 56² = 65²
1089 + 3136 = 4225
4225 = 4225
Example: a = 6 and b = 4
a² + b² = 36 + 16 = 52
a² – b² = 36 – 16 = 20
2ab = 2(24) = 48 20² + 48² = 52²
400 + 2304 = 2704
2704 = 2704
also if 20 – 40 – 52 divide by 4
5 – 12 – 13
5² + 12² = 13² (our 2nd basic odd triple)
25 + 144 = 169
169 = 169
Try some on your own and find some more of those even numbered one
like 20 – 21 – 29.
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