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The Legendre symbol tells us to calculate $5^{350} \mod 701$, but this question was on an exam where no calculators are allowed, so I wasn't able to do this question. How can you find if $5$ is a square $\mod 701$ without a calculator if $701$ is prime? What if it's not a prime number?

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Use the law of quadratic reciprocity.

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    $\begingroup$ Since $5\equiv 1(4)$, 5 is a quadratic residue $\mod 701$ iff $701$ is a quadratic residue $\mod 5$. Can you check if 701 is a quadratic residue $\mod 5$? $\endgroup$ – LASV Mar 24 '15 at 20:15
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    $\begingroup$ The formula you put above my first comment answers your question: Let $p=5, q=701$. Since $5$ is congruent to 1 mod 4, $p-1/2$ is even and hence the right side is equal to one. Now you know ${5 \choose 701} {701 \choose 5}=1$. But ${701 \choose 5}$ is easy to compute, right? $\endgroup$ – LASV Mar 24 '15 at 20:24
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    $\begingroup$ The Legendre symbol is only for odd primes, correct? An odd prime is of the type $p\equiv 1 \mod 4$ or $p\equiv 3 \mod 4$. If $p\equiv 1 \mod 4$, the exponent on the right side is even, hence you get a value of 1 (Is this clear? $p-1$ is divisible by 4). $\endgroup$ – LASV Mar 24 '15 at 20:30
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    $\begingroup$ The law of quadratic reciprocity splits the behavior of the primes in 2 categories: 1 mod 4 and 3 mod 4. $\endgroup$ – LASV Mar 24 '15 at 20:31
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    $\begingroup$ The sign on the right side is still positive if either prime is congruent to 1 mod 4. $\endgroup$ – LASV Mar 25 '15 at 2:45
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Is there anyway to solve this using Fermat's little theorem? a^(p-1) ≡ 1 mod p.

5^700 ≡ 1 mod 701

(5^350)^2 ≡ 1 mod 701

But I'm not sure if this tells us anything about 5^350 mod 701

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