The Pythagorean Triple Truth

Truth or dare? I dare you to find a Pythagorean triple that includes the number 314159265. Not so easy, is it? You want a truth? Here it is:

• Take any odd number, a for example.
• Square it.
• Divide by 2.
• Subtract and add ½.
• These numbers are your b and c!

Simplified form:
b = a²/2 - ½
c = a²/2 + ½


So lets try it!
a = 314159265
b = (314159265)²/2 - ½ = 49,348,021,892,670,112
c = (314159265)²/2 + ½ = 49,348,021,892,670,113

If it works for Wolfram Alpha, it works for me!

Now, if we plug in these values of a into the Pythagorean theorem, something expected happens:

(a²/2+½)²-(a²/2-½)²-a²

(a²+1)²/4-(a²-1)²/4-a²

(a^4+2a²+1)/4-(a^4-2a²+1)/4-a²

(4a²)/4-a² = 0

Wait a second, shouldn't this work for any number a? It does, but if an even number is used then the result technically isn't a Pythagorean triple, since b and c are fractions.


The formula will always find a Pythagorean triple where b and c have a difference of 1. I derived it when I noticed that if c-b=1, then c²-b²=b+c. Since b and c are consecutive numbers, b+c will always be odd. If b+c is also a perfect square, then the result will be a Pythagorean triple. The easiest way to make an odd perfect square is to take any odd number a and square it. Dividing by two and subtracting and adding ½ will find the two consecutive numbers b and c. And that is my Pythagorean triple truth!


Grant