
Glossary: Prime Pages: Top 5000: 
Suppose p is an odd prime and a is any
integer. The Legendre symbol (ap) is defined to be
Note: the Legendre symbol is better written vertically: , but this is difficult and slow on web pages. Euler showed that (ap) = a^{ (p1)/2} (mod p). Using this we can show the following: Let p and q be odd primes, then (1p) = 1 if p = 1 (mod 4), and (1p) = 1 if p = 3 (mod 4);For the prime 2 we have (2p) = 1 if p = 1 or 7 (mod 8), andFar more difficult to prove is the quadratic reciprocity law: If p and q are distinct primes, then .In other words, (pq) = (qp), unless p = q = 3 (mod 4), in which case (pq) = (qp). The Legendre symbol is often evaluated by using the Jacobi symbol.
See Also: JacobiSymbol
Chris K. Caldwell © 19992014 (all rights reserved)
