3
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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.

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4
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$$100 003\equiv3\pmod4$$

For any integer $\displaystyle c\equiv0,1,2,3\pmod4\implies c^2\equiv0,1 $

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  • $\begingroup$ I'm not sure I understand. This is not in base 4. $\endgroup$ – Andrei Jun 6 '14 at 12:10
  • $\begingroup$ Okay now I got thank you very much. $\endgroup$ – Andrei Jun 6 '14 at 12:16
  • $\begingroup$ @Andrei, Nice to hear that $\endgroup$ – lab bhattacharjee Jun 6 '14 at 12:20
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    $\begingroup$ I must say, I for one could use a little more explanation. $\endgroup$ – Bennett Gardiner Jun 6 '14 at 12:35
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    $\begingroup$ @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. $\endgroup$ – mursalin Jun 6 '14 at 13:52

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