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Exercise: Solve the following equation:
x2
3 (mod 7)
We attempt to use the explicit formula for computing roots of congruence classes modulo a prime:
x
± (3)(7+1)/4 (mod 7)
± 2 (mod 7)
(2)2
3 (mod 7)
Since the formula did not return a valid root, there is no solution; 3 is not a quadratic residue in ℤ/7ℤ