For any integer a and any positive odd integer n, the Jacobi symbol (⁠a/n⁠) is defined as the product of the Legendre symbols corresponding … See more

The following facts, even the reciprocity laws, are straightforward deductions from the definition of the Jacobi symbol and the corresponding properties of the Legendre symbol.
The Jacobi … See more

The above formulas lead to an efficient O(log a log b) algorithm for calculating the Jacobi symbol, analogous to the Euclidean algorithm for finding the gcd of two numbers. (This should not be surprising in light of rule … See more

The Legendre symbol (⁠a/p⁠) is only defined for odd primes p. It obeys the same rules as the Jacobi symbol (i.e., reciprocity and the supplementary formulas for (⁠−1/p⁠) and (⁠2/p⁠) and multiplicativity of the "numerator".)
Problem: G… See more

There is another way the Jacobi and Legendre symbols differ. If the Euler's criterion formula is used modulo a composite number, the result may or may not be the value of the Jacobi symbol, and in fact may … See more

• Kronecker symbol, a generalization of the Jacobi symbol to all integers.
• Power residue symbol, a generalization of the Jacobi symbol to higher powers residues. See more