1. $[\mathbf{3}]$ If $a$ and $b$ are positive integers such that $a^{2}-b^{4}=2009$, find $a+b$.
Answer:
47
Solution: We can factor the equation as $\left(a-b^{2}\right)\left(a+b^{2}\right)=41 \cdot 49$, from which it is evident that $a=45$ and $b=2$ is a possible solution. By examining the factors of 2009 , one can see that there are no other solutions.