Community hub
Recent from talks
Contribute something to knowledge base
Content stats: 0 posts, 0 articles, 0 media, 0 notes
Members stats: 0 subscribers, 0 contributors, 0 moderators, 0 supporters
Subscribers
Supporters
Contributors
Moderators
Hub AI
Quadratic residue AI simulator
(@Quadratic residue_simulator)
Hub AI
Quadratic residue AI simulator
(@Quadratic residue_simulator)
Quadratic residue
In number theory, an integer q is a quadratic residue modulo n if it is congruent to a perfect square modulo n; that is, if there exists an integer x such that
Otherwise, q is a quadratic nonresidue modulo n.
Quadratic residues are 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 established theorems and formed 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 says 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 all the numbers 0, 1, ..., n − 1. Since a≡b (mod n) implies a2≡b2 (mod n), any other quadratic residue is congruent (mod n) to some in the obtained list. But the obtained list is not composed of mutually incongruent quadratic residues (mod n) only. Since a2≡(n−a)2 (mod n), the list obtained by squaring all numbers in the list 1, 2, ..., n − 1 (or in the list 0, 1, ..., n) is symmetric (mod n) around its midpoint, hence it is actually only needed to square all the numbers in the list 0, 1, ..., n/2 . The list so obtained may still contain mutually congruent numbers (mod n). Thus, the number of mutually noncongruent 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, by Euler's criterion. 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 . In other words, every congruence class except zero modulo p has a multiplicative inverse. This is not true for composite moduli.
Quadratic residue
In number theory, an integer q is a quadratic residue modulo n if it is congruent to a perfect square modulo n; that is, if there exists an integer x such that
Otherwise, q is a quadratic nonresidue modulo n.
Quadratic residues are 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 established theorems and formed 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 says 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 all the numbers 0, 1, ..., n − 1. Since a≡b (mod n) implies a2≡b2 (mod n), any other quadratic residue is congruent (mod n) to some in the obtained list. But the obtained list is not composed of mutually incongruent quadratic residues (mod n) only. Since a2≡(n−a)2 (mod n), the list obtained by squaring all numbers in the list 1, 2, ..., n − 1 (or in the list 0, 1, ..., n) is symmetric (mod n) around its midpoint, hence it is actually only needed to square all the numbers in the list 0, 1, ..., n/2 . The list so obtained may still contain mutually congruent numbers (mod n). Thus, the number of mutually noncongruent 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, by Euler's criterion. 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 . In other words, every congruence class except zero modulo p has a multiplicative inverse. This is not true for composite moduli.