<s>[INST] Find the smallest positive integer which can be written as the sum of the squares of two positive integers in two different ways 
Find the smallest positive integer which can be written as the sum of the squares of two positive integers in two different ways.

I took extremely long to solve this 
I got 
$50= 7^2 + 1^2 $
$50= 5^2 + 5 ^2 $ 
I did it with the trial and error method. 
I'm just curious if this came out during my exams, is there a quicker way to find the answer?
 [/INST] I do wonder why it took so long... I added all pairs from the first ten squares on a piece of scrap paper in a few minutes. Maybe that's a long time. 
\begin{array}{c|c|c}
&& +1 & +4 & +9 & +16 &  +25 & +36 & +49 & +64 & +81 & +100  \\ \hline
1 && 2 \\ \hline
4 && 5 & 8\\ \hline
9 && 10 & 13 & 18 \\ \hline
16 && 17 & 20 & 25 & 32 \\ \hline
25 && 26 & 29 & 34 & 41 & \color{red}{50} \\ \hline
36 && 37 & 40 & 45 & 52 & 61 & 72 \\ \hline
49 && \color{red}{50} & 53 & 58 & \color{blue}{65} & 74 & \color{magenta}{85} & 98 \\ \hline
64 && \color{blue}{65} & 68 & 73 & 80 & 89 & 100 & 113 & 128 \\ \hline
81 && 82 & \color{magenta}{85} & 90 & 97 & 106 & 117 & 130 & 145 & 162 \\ \hline
100 && 101 & 104 & 109 & 116 & 125 & 136 & 149 & 164 & 181 & 200 \\ \hline
\end{array}
This also gives the first case when all the squares in question are distinct, at $65 = 8^2+1^2 = 7^2+4^2$
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