Legendre symbol (a/p): 1 if a≡x² (mod p) for some x (quadratic residue), −1 if not (non-residue), 0 if p|a. Euler's criterion: (a/p) ≡ a^((p−1)/2) (mod p). Quadratic reciprocity: (p/q)(q/p) = (−1)^((p−1)/2·(q−1)/2).
How to Use
Enter a and prime p:
- (a/p): 1, −1, or 0
- Euler: a^((p−1)/2) mod p
- QR list: All residues mod p
Quadratic Reciprocity
For odd primes p≠q: (p/q)(q/p) = (−1)^((p−1)(q−1)/4). Supplements: (−1/p) = (−1)^((p−1)/2), (2/p) = (−1)^((p²−1)/8). Gauss called this the 'golden theorem.'
Applications
- Solving quadratic congruences
- Primality testing
- Cryptography (Goldwasser-Micali)
- Algebraic number theory
Step-by-Step Instructions
- 1Enter a.
- 2Enter prime p.
- 3Get (a/p).
- 4View Euler computation.
- 5See QR list.