What is the least quadratic residue mod p?

The least quadratic residue mod p is clearly 1. The question of the magnitude of the least quadratic non-residue n(p) is more subtle, but it is always prime. The Pólya–Vinogradov inequality above gives O(√p log p).

How do you find the number of quadratic residues modulo n?

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, by Euler’s criterion.

What are quadratic residues and how are they used?

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 .

What is the product of a residue and a nonresidue?

The product of a residue and a nonresidue may be a residue, a nonresidue, or zero. For example, from the table for modulus 6 1, 2, 3, 4, 5 (residues in bold ). The product of the residue 3 and the nonresidue 5 is the residue 3, whereas the product of the residue 4 and the nonresidue 2 is the nonresidue 2.

What is the maximum number of quadratic residues modulo n?

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.

What is the difference between a residue and a non residue?

Following this convention, modulo an odd prime number there are an equal number of residues and nonresidues. Modulo a prime, the product of two nonresidues is a residue and the product of a nonresidue and a (nonzero) residue is a nonresidue.

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