Quadratic Residue - History, Conventions, and Elementary Facts

History, Conventions, and Elementary Facts

Fermat, Euler, Lagrange, Legendre, and other number theorists of the 17th and 18th centuries proved some theorems and made some conjectures about quadratic residues, but the first systematic treatment is § IV of Gauss's Disquisitiones Arithmeticae (1801). Article 95 introduces the terminology "quadratic residue" and "quadratic nonresidue", and states that, if the context makes it clear, the adjective "quadratic" may be dropped.

For a given n a list of the quadratic residues modulo n may be obtained by simply squaring the numbers 0, 1, …, n − 1. Because a2 ≡ (n − a)2 (mod n), the list of squares modulo n is symmetrical around n/2, and the list only needs to go that high. This can be seen in the table at the end of the article.

Thus, the number of quadratic residues modulo n cannot exceed n/2 + 1 (n even) or (n + 1)/2 (n odd).

The product of two residues is always a residue.