An example I can see using basic algebra is $\sqrt{2} + \sqrt{8} = \sqrt{18}$, but is there a general method to find integer solutions to the problem? Another question: say you are given the value of $c$; can you find the values of $a$ and $b$?
Integer solutions to $\sqrt{a} + \sqrt{b} = \sqrt{c}$
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number-theory
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3As a starting point, if $(\sqrt a+\sqrt b)^2\in\Bbb Z$, then $ab$ must be a square number, as $(\sqrt a+\sqrt b)^2=a+b+2\sqrt{ab}$. – 2012-12-01
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1Simplify by dividing out the least common divisor. Is it true that what remains will be integer square-roots only? In this example, divide out by 2, and get $\sqrt1+\sqrt4=\sqrt9$ – 2012-12-01
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0Other solutions are $a=0,\ b=c$ and $a=c,\ b=0$, and these are all of the solutions with one of the unknowns equal to zero. Also, they all have to be of the same sign if none of them is zero, and the negative solutions are in a bijective correspondence with the positive ones. So you can only consider $a,b,c>0$. – 2012-12-01