
The Jacobi symbol is an extension of the Legendre symbol to any odd modulus, using the rule (a/bc) = (a/b)(a/c) to decompose the modulus as a product of primes.
If a is an integer and n>0 is odd, then  (  a n 
)  =  (  a p_{1} 
)  e_{1} 
(  a p_{2} 
)  e_{2} 
...  (  a p_{k} 
)  e_{k} 
, where 
p_{1}^{e}1, p_{2}^{e}2, ..., p_{k}^{e}k is the prime factorization of n, and (a/p_{i}) are Legendre Symbols.
Alternatively,
If a is an integer and n>0 is odd, then  (  a n 
)  =  (  a p_{1} 
)  (  a p_{2} 
)  ...  (  a p_{k} 
)  , where 
p_{1}, p_{2}, ..., p_{k} are all the primes dividing n/s, where s is the largest square factor of n, and (a/p_{i}) are Legendre Symbols.
Still another formulation is
If a is an integer and n>0 is odd, then  (  a n 
)  is  { 
0 if GCD(a,n)≠1, 
A consequence of this last formulation of the Jacobi Symbol is that if (a/n)=1, then a is for sure not a quadratic residue (mod n); however, if (a/n)=1, then a may or may not be a quadratic residue (mod n). Examples of this phenomenon include:
(2/3)=1 and (2/5)=1, so 2 is not a quadratic residue (mod 15), yet (2/15)=1.
(2/7)=1 and (2/17)=1, so 2 is a quadratic residue (mod 119), and (2/119)=1.
So in these two cases, (2/15)=1 and (2/119)=1, so if the (a/n)=1, then a may or may not be a quadratic residue (mod n).
If a,n are not coprime, then Jacobi (a/n)=0, and, again, a may or may not be a quadratic residue (mod n).
Mathworld: Jacobi Symbol
Prime Glossary: Jacobi Symbol
Wikipedia: Jacobi symbol
Jacobi Symbol Algorithm  programs written in pseudocode, Javascript, VBA (Microsoft Visual Basic for Applications)
Legendre symbol — ( a
p) , where a is any integer, and p is an odd prime
Kronecker symbol — ( a
n) , where a and n are any integers, an extension of the Jacobi symbol.
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