Modular square root
In number theory, an integer q is called a quadratic residue modulo n if it is congruent to a perfect square modulo n; i.e., if there exists an integer x such that:
Otherwise, q is called a quadratic nonresidue modulo n.
Originally an abstract mathematical concept from the branch of number theory known as modular arithmetic, quadratic residues are now used in applications ranging from acoustical engineering to cryptography and the factoring of large numbers.
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 below.
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.
Modulo 2, every integer is a quadratic residue.
Modulo an odd prime number p there are (p + 1)/2 residues (including 0) and (p − 1)/2 nonresidues. In this case, it is customary to consider 0 as a special case and work within the multiplicative group of nonzero elements of the field Z/pZ. (In other words, every congruence class except zero modulo p has a multiplicative inverse. This is not true for composite moduli.)
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