Exercise 3.7.7

Question

Let $p = 2437$. Use the Euler criterion to compute $R^*(5,p)$ and $R^*(28,p)$.

Richard Ganaye · Accepted Answer

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\begin{proof} By Proposition 3.4.5, since $p = 2437$ is prime,
$$R(5,p) = 1 + \legendre{\Delta}{p} = 1 + \legendre{5}{2437}.$$
By fast exponentiation,
$$\legendre{5}{2437} \equiv 5^{1218} \equiv - 1 \pmod {2437}.$$
Therefore $R(5,p) = R^*(5,p) = 0$.

Alternatively, using quadratic reciprocity, we obtain
$$ \legendre{5}{2437}  = \legendre{2437}{5} = \legendre{2}{5} = -1.$$

\bigskip
Similarly, $\legendre{28}{2437}  = 1$. Using Proposition 3.4.5, 
$$R^*(28,p) = R(28,p) = 1 + \legendre{28}{2437} = 2.$$

Exercise 3.7.7

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Exercise 3.7.7

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