Jacobi symbol (a/n): for odd n = p₁^e₁·...·pₖ^eₖ, (a/n) = Π(a/pᵢ)^eᵢ. Generalizes Legendre symbol to composite moduli. (a/n)=1 doesn't guarantee a is a QR mod n. Used in Solovay-Strassen primality test.
How to Use
Enter a and odd n:
- (a/n): Product of Legendre symbols
- Factors: Factorization of n
- Steps: Evaluation process
Properties
- (a/n) = 0 iff gcd(a,n)>1
- (ab/n) = (a/n)(b/n)
- (a/mn) = (a/m)(a/n)
- Reciprocity still holds
Algorithm
Efficient computation without factoring n: use reciprocity, (−1/n), (2/n) supplements, and reduction mod n. Runs in O(log²n) — same as GCD.
Step-by-Step Instructions
- 1Enter a.
- 2Enter odd n.
- 3Get (a/n).
- 4View steps.
- 5See factorization.