$a + b = c + c$ So I have the following problem: $a + b = c + c.
I want to prove that the equation has infinitely many relatively prime integer solutions.
What I did first was factor the right side to get: 
(

Suppose $c=u^2+v^2$ then you have $$(uc^2+v)^2+(vc^2-u)^2=c(c^4+1)$$ and you want $uc^2+v$ and $vc^2-u$ to be coprime.
Now just choose $u=1$ so that the expressions are $a=c^2+v$ and $b=vc^2-1$ with $c=v^2+1$
Note then that that a common factor of $a$ and $b$ is also a factor of $va-b=v^2+1=c$ and $ac^2-b=c^4+1$, but $c$ and $c^4+1$ are coprime.

So you get a family of solutions with 
$c=v^2+1$ and 
$a=c^2+v=v^4+2v^2+v+1$ and 
$b=vc^2-1=v^5+2v^3+v-1$
[Which, I notice is just Sam's parametric solution with $u=1$. It should be obvious what to do to generalise.]
