# sum of 2 squared integers

Let $$a^2 + b^2 = 100003,$$ are there any integers $a$ and $b$ where this is true? I have tried to figure out the individual digits for $a$ and $b$ to be true. I figured out that the sum of the last one has to $13$ and as such one integer has the last digit $9$ and the other one $4$, next their nest sum is $9$ but I have not found any pattern in the second digit.

$$100 003\equiv3\pmod4$$
For any integer $\displaystyle c\equiv0,1,2,3\pmod4\implies c^2\equiv0,1$
• @BennettGardiner, a perfect square is always $\equiv 0, 1\pmod 4$. So, $a^2+b^2\equiv 0, 1, 2\pmod 4$, but the right hand side is $\equiv 3\pmod 4$. So the equality can not hold. – mursalin Jun 6 '14 at 13:52