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I have been stuck with a question on eliptic curves lately. I need to know whether perfect square mod n is different than a normal perfect square.

And also is 3 a perfect square in mod 13?

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    $\begingroup$ You need to look into quadratic residues. Yes, 3 is a perfect square (i.e. a quadratic residue) mod 13, since $4^2 \equiv 16 \equiv 3 \pmod{13}$. $\endgroup$
    – Old John
    Oct 24, 2013 at 10:46
  • $\begingroup$ Do you see that, $3$ in some sense same as $4^2 \text{mod} 13$.. $\endgroup$
    – user87543
    Oct 24, 2013 at 10:46

2 Answers 2

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Yes, 3 is a perfect square $\bmod 13$ because $4^2 \equiv 16 \equiv 3 \bmod 13.$ All normal squares (i.e. 1,4, 9) less than 13 obviously are perfect squares $\bmod 13$, but as the example 3 shows there are more than these.

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$x$ is said to be a perfect square in modulo $n$ if $\exists y$ such that $ y^2\equiv x \pmod{n} $.

And yes, $3$ is a perfect square in$\pmod{13}$ because $4^2\equiv 3\pmod{13}$.

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