The Jacobi Symbol

Let  
\[n, \; m\]
  be odd integers with  
\[n \gt 3, \; gcd(m, \; n)=1\]
. Write  
\[n=p_1p_2...p_r\]
  where the  
\[p_i\]
  are not necessarily distinct.>br /> The Jacobi symbol is defined as  
\[(m/n)=(n/p_1)(n/p_2)...(n/p_r)\]
  where each factor on the right hand side is a Legendre Symbol.
For example,
\[(14/275)=(14/5)(14/5)(14/11)=(14/5)(14/5)(3/11)==1 \times 1 \times 1\]

since  
\[1\]
  is a Quadratic Residue of 5 and  
\[3 \equiv 25 \; (mod \; 11) \equiv 5^2 \; (mod \; 11)\]
  so 3 is a quadratic residue of 11.