Evaluate (a/p) and test quadratic residue status modulo an odd prime
a
odd prime p
Result
Answer
Step-by-Step Derivation
Legendre Symbol Formula
(a/p) = 1 if a is a quadratic residue mod p
(a/p) = -1 if a is a non-residue mod p
Euler criterion: (a/p) ≡ a^((p-1)/2) (mod p)
The Legendre symbol compresses quadratic residue information into one value. Euler criterion gives a practical calculation by raising a to half of p minus one modulo the odd prime p.
⚠The modulus p must be an odd prime. For composite moduli, use Jacobi symbol methods instead of the Legendre symbol.
What Is the Legendre Symbol?
The Legendre symbol is a number theory notation that tells whether a number is a square modulo an odd prime. It is closely related to modular square roots and quadratic reciprocity.
Residue Result
Value 1 means x^2 = a mod p has a solution.
Non-Residue Result
Value -1 means no square root exists modulo p.
Zero Case
Value 0 means p divides a.
Euler Criterion
A fast exponent test evaluates the symbol.
Example: For a = 5 and p = 11, compute 5^5 mod 11. The result determines whether 5 is a quadratic residue modulo 11.
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