Simple Equation « Thoughts on my Mind

Simple Equation December 2, 2008

Posted by putnam120 in Uncategorized.
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Well we all know that the only solutions to $x^2+y^2=(x+y)^2$ are $(0,r)$ and $(r,0)$ where $r\in\mathbb{R}$ . So let us move onto something a little more “difficult”. Find all integer triplets satisfying the following equation.

$a^3+b^3+c^3=(a+b+c)^3$

We begin by doing the obvious thing and expand the right hand side and then cancel terms, thus leaving us with $3ab(a+b)+3ac(a+c)+3bc(b+c)+6abc=0$ . Then divide by 3 to get $ab(a+b)+ac(a+c)+bc(b+c)+2abc=0$ , which after some simple regrouping factors into $(a+b)(a+c)(b+c)=0$ . From here it is obvious that the only solutions are those where one of $a,b,c$ is euqal to the negation of one of the others. See pretty simple, the only real work is being able to expand the first expression. The factoring might be a little tricky but just from looking at the equation you should have a general idea of what the solution should be and that should help you get pretty far with the factoring.