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Law of quadratic reciprocity

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The law of quadratic reciprocity predicts whether an odd prime number [math]\displaystyle{ p }[/math] is a quadratic residue or non-residue modulo another odd prime number [math]\displaystyle{ q }[/math] if we know whether [math]\displaystyle{ q }[/math] is a quadratic residue or non-residue modulo [math]\displaystyle{ p }[/math].

  • If at least one of [math]\displaystyle{ p }[/math] or [math]\displaystyle{ q }[/math] are congruent to 1 mod 4: [math]\displaystyle{ p }[/math] is a quadratic residue modulo [math]\displaystyle{ q }[/math] if and only if [math]\displaystyle{ q }[/math] is a quadratic residue modulo [math]\displaystyle{ p }[/math].
  • If both of [math]\displaystyle{ p }[/math] or [math]\displaystyle{ q }[/math] are congruent to 3 mod 4: [math]\displaystyle{ p }[/math] is a quadratic residue modulo [math]\displaystyle{ q }[/math] if and only if [math]\displaystyle{ q }[/math] is a quadratic non-residue modulo [math]\displaystyle{ p }[/math].

This theorem was first proved by Carl Friedrich Gauss in 1801.

This does not cover the cases where we want to know whether -1 or 2 are quadratic residues or non-residues modulo [math]\displaystyle{ p }[/math].

  • 2 is a quadratic residue modulo [math]\displaystyle{ p }[/math] if and only if [math]\displaystyle{ p }[/math] is congruent to 1 or 7 (mod 8).
  • -1 is a quadratic residue modulo [math]\displaystyle{ p }[/math] if and only if [math]\displaystyle{ p }[/math] is congruent to 1 (mod 4).

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