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though this one may seem a duplicate of - say - this: Determine all primes $p$ for which $5$ is a quadratic residue modulo $p$, well, I have a question. How would you manage the problem of determining the set of odd primes for which a quadratic residue modulo $p$, when $p$ is 'large'. For instance, suppose $p=29$. Then, one applies the quadratic reciprocity law obtaining: $$ \left(\frac{29}{p}\right)=\left(\frac{p}{29}\right)\cdot\left(-1\right)^{\left(p-1\right)\left(29-1\right)/4} $$

So we need to study the following:

  1. the sign of the Legendre symbol, so the values of $p$ modulo $29$
  2. the parity of $\left(-1\right)^{\left(p-1\right)/2}$ modulo 2, or the parity of $p$ modulo $4$

Now, this suggests to consider values of $p$ modulo $29\cdot4=116$ and since $\varphi\left(116\right)=56$, this approach is surely exhaustive, though there must be a more clever, and faster way. Any suggestion?

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Since the final answer will necessarily look like "$29$ is a quadratic residue modulo $p$ if and only if $p\mod 116$ is one of the following values ..." a more clever and faster way can hardly be imagined.

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