Discussion
Which of the following is a quadratic residue modulo 43 ?
Answer: B
The quadratic residue is 40.
\(40 \equiv -3 \pmod{43}\). Using Legendre symbols, with \(43 \equiv 3 \pmod 4\):
\[ \begin{aligned} \left(\tfrac{-1}{43}\right) &= -1 \\ \left(\tfrac{3}{43}\right) &= -\left(\tfrac{43}{3}\right) = -\left(\tfrac{1}{3}\right) = -1 \\ \left(\tfrac{-3}{43}\right) &= (-1)(-1) = 1 \end{aligned} \]
So 40 is a residue. (Indeed \(13^{2} = 169 = 3 \cdot 43 + 40\).) The others are not: -1 and 3 have symbol -1 as computed, and since \(43 \equiv 3 \pmod 8\), \((2/43) = -1\).
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