Calculator Input
Example Data Table
| a | n | Jacobi Symbol (a/n) | Quick Note |
|---|---|---|---|
| 5 | 7 | -1 | Prime modulus example with a nonresidue. |
| 2 | 15 | 1 | Composite modulus example using reciprocity rules. |
| 5 | 21 | 1 | Composite modulus can still return positive one. |
| 10 | 21 | -1 | Useful for checking factor parity behavior. |
| 3 | 9 | 0 | Zero appears when numerator shares factors. |
Formula Used
1. The Jacobi symbol is defined for odd positive n:
(a/n) = Π (a/pi)ei, where n = Π piei.
2. Reduce the numerator first:
(a/n) = (a mod n / n)
3. Remove factors of two with the parity rule:
(2/n) = (-1)(n² - 1)/8
4. Apply quadratic reciprocity for odd values:
(a/n) = (-1)((a - 1)(n - 1))/4(n mod a / a)
This calculator uses repeated reduction, factor-two extraction, and reciprocity sign changes until the numerator becomes zero or the modulus becomes one.
How to Use This Calculator
- Enter any integer numerator
a. - Enter an odd positive modulus
n > 1. - Set graph start, end, and step values.
- Choose a bar chart or line chart.
- Press Calculate Jacobi Symbol.
- Review the final symbol above the form.
- Check the detailed reciprocity steps and interpretation.
- Download CSV or PDF for records and study.
Frequently Asked Questions
1. What does the Jacobi symbol measure?
It generalizes the Legendre symbol to any odd positive modulus. It summarizes quadratic residue behavior across prime factors of the modulus, but for composite moduli it does not always confirm solvability of x² ≡ a mod n.
2. What inputs are valid here?
Use any integer for a and any odd positive integer greater than one for n. Even or nonpositive moduli are invalid because the Jacobi symbol is defined only for odd positive moduli.
3. What does a result of 0 mean?
A zero result means the numerator and modulus share a nontrivial common factor. In practice, gcd(a, n) is greater than one, so the symbol immediately signals shared divisibility.
4. Is a result of 1 always a quadratic residue?
No. For prime modulus, yes. For composite modulus, a result of 1 only reflects the product of Legendre symbols across prime factors. The congruence x² ≡ a mod n may still have no solution.
5. How is this different from the Legendre symbol?
The Legendre symbol requires an odd prime modulus. The Jacobi symbol allows any odd positive modulus. When n is prime, both symbols agree exactly.
6. Why does the calculator show steps?
The step list shows each normalization, factor-of-two extraction, reciprocity sign flip, and swap-reduce move. This makes the algorithm easier to verify, study, and reuse by hand.
7. What is the graph showing?
The graph plots values of (k/n) over your selected residue range. It helps reveal repeating patterns, zeros, and the balance between positive and negative symbol values.
8. Can I save the result?
Yes. Use the CSV button for structured data and graph points. Use the PDF button for a clean result snapshot that includes the computed symbol, summary values, and steps.