Shorten equations with math notations?

[BadatPhysicsguy]

Homework Statement

"Show that the three smallest pythagorean triangles are (3, 4, 5), (6, 8, 10) and (5,12, 13).

Homework Equations

a^2 + b^2 = c^2

The Attempt at a Solution

The obvious thing to do would be to simply show that a triangle with the hypothenuse 1, 2, 3, 4, 6, 7, 8, 9, 11 and 12 simply do not satisfy the pythagorean equation (a^2+b^2=c^2) but instead of showing this through 11 separate solutions, is there anyway to shorten this, generalize it or use something like a sum notation (sigma)? Would be pretty unneccesary to waste two pages on simply writing it all out.

 

[ShayanJ]

You can write a computer program to do that for you. It won't be hard!

 

[haruspex]

First question is what is meant by smallest here? The most reasonable is by area, so it might not be ok only to look at hypotenuses less than 13.
Assuming area, better to work in terms of the length of the shortest side - fewer choices.
It is possible to deduce a general form for solutions. Not sure if you're expected to be able to do that.

 

[Joffan]

I don't know about shortening anything, but it does clarify to put some mathematical notation on the problem. So for example say the triangle edge lengths are labelled $a$, $b$ and (hypotenuse) $c$, with $0<a \leq b<c$.

Then you can review possibilities based on $a$ and $b$. For a particular $a$, you could infer where to stop with $b$ by finding the smallest $n$ such that $(n+1)^2 - n^2 = 2n + 1 > a^2$

Also, you could investigate square numbers under modular arithmetic. This would let you show that $a$ and $b$ needs at least one multiple of 3 and one even number (mod 4 tells you this) and (interestingly) that either $a$, $b$, $(a+b)$ or $(b-a)$ must be a multiple of 7.

Of course you could just draw up an exhaustive table too. Sometimes brute force is quicker, especially in Excel.