
Glossary: Prime Pages: Top 5000: 
In the study of diophantine equations (and surprisingly
often in the study of primes) it is important to know
whether the integer a is the square of an integer
modulo p. If it is, we say a is a
quadratic residue modulo p; otherwise, it is
a quadratic nonresidue modulo p. For
example, 4^{2}=7 (mod 9) so 7 is a quadratic
residue modulo 9. Lets look at a few more examples:
For an odd prime p, there are (p+1)/2 quadratic residues (counting zero) and (p1)/2 nonresidues. (The residues come from the numbers 0^{2}, 1^{2}, 2^{2}, ... , {(p1)/2}^{2}, these are all different modulo p and clearly list all possible squares modulo p.) When the base is a product of odd prime powers, and the numbers in question are relatively prime to the base, then
See Also: LegendreSymbol, JacobiSymbol
Chris K. Caldwell © 19992018 (all rights reserved)
