Is there anything remarkable about the fact that 3^{2} + 4^{2} = 5^{2}?
A couple of random mathematical conversations occurred at work the other week, during one of which the above equation was mentioned (it's a commonly known integer solution for a right angle triangle). It was noted that some people find it too coincidental that things like the above are true. Personally, I'm of the reverse opinion. There's a million other equations we could test for small numbers that wouldn't turn out to be true, it's just this one is. It might seem interesting, but is there really anything unique about it...?
Let's have a meandering chat about some mathematical concepts:
There's An Infinite Number Of Numbers
This is a pretty trivial statement that most people come across early in their school days. The proof? It's simply that if you think there's a "biggest number" we can always add 1 to it to make an even bigger one. For ever. Not much to say here, but let's throw around some basic stuff first.
The Sum Of The First N Odd Numbers Is A Square
This is another one that's relatively simple to demonstrate:
1 = 1^{2} = 1 4 = 2^{2} = 1 + 3 9 = 3^{2} = 1 + 3 + 5 16 = 4^{2} = 1 + 3 + 5 + 7 25 = 5^{2} = 1 + 3 + 5 + 7 + 9 36 = 6^{2} = 1 + 3 + 5 + 7 + 9 + 11 49 = 7^{2} = 1 + 3 + 5 + 7 + 9 + 11 + 13 ... and so on.
Basically, start from 1 and add the next odd number, and the next, and the next, and... Do this as many times as you like and you will always end up with a square number. Further, for any odd number there's some square number which if you add that odd number to it will also result in a square (e.g. for odd number 13, 7^{2} = 6^{2} + 13).
Combine The Two
So, what can get from the two topics above?
Well, from the first point we know there's obviously an infinite amount of odd numbers. Let's call one of those numbers X. X^{2} is also an odd number.
And from our second point, we know there's always some square number Y^{2} such that the (Y+1)^{2} = Y^{2} + X^{2}.
With a bit of magic maths, we can get a bit further than this too. I'll leave it as an exercise for the reader, but with some number crunching we can relatively easily determine that for a given X in the above equation, Y = (X^{2}  1) / 2.
Let's start calculating some of these:
5^{2} = 4^{2} + 3^{2} 13^{2} = 12^{2} + 5^{2} 25^{2} = 24^{2} + 7^{2} 41^{2} = 40^{2} + 9^{2} 61^{2} = 60^{2} + 11^{2} 85^{2} = 84^{2} + 13^{2} ... and so on.
So, basically, our original formula is just one of an infinite number of "right angled triangle" solutions, all following a regular pattern where the two longest sides differ by just one unit.
In Summary
You might still feel that 3/4/5 is still worthy of consideration as being a special and unique case though. After all, no other solutions form a sequence of three numbers, right? And it's the first solution for the equation?
Well, maybe... If it weren't for that fact that 1^{2} = 0^{2} + (1)^{2}.
Recommended further reading: Fermat's Last Theorem
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