Currently there may be errors shown on top of a page, because of a missing Wiki update (PHP version and extension DPL3).
Navigation
Topics Help • Register • News • History • How to • Sequences statistics • Template prototypes

Law of quadratic reciprocity

From Prime-Wiki
Revision as of 22:45, 26 January 2019 by Karbon (talk | contribs) (restored)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

The law of quadratic reciprocity predicts whether an odd prime number p 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).

External links