Kronecker Symbol
Extend the Jacobi symbol to signed decimal denominators, including factors of two, negative denominators, and (a/0)=1 exactly when |a|=1.
Description
The Kronecker symbol extends the Jacobi symbol to even, negative and zero denominators.
The Kronecker symbol extends the Jacobi symbol to even, negative and zero denominators. Special sign and power-of-two rules make its domain broader than quadratic-residue tests modulo an odd prime.1
The tool returns symbol. The indexing and output conventions below determine how to read these values.
When to use Kronecker Symbol
Use this for the following task: Extend the Jacobi symbol to signed decimal denominators, including factors of two, negative denominators, and (a/0)=1 exactly when |a|=1. Use the input and output conventions below to reproduce the intended mathematical quantity.
How to use Kronecker Symbol and understand its results
- Choose the required indexes, integers or sequence entries. Preserve decimal-string encodings where the field table specifies strings.
- Enter numerator, denominator according to the field table, then run the tool.
- Read symbol. Use the stated units, indexing and interpretation of the numerical result.
- Keep exact decimal strings for subsequent integer arithmetic; distinguish them from any explicitly approximate numeric result.
| Input field | Meaning | Value type and bounds | Requirement |
|---|---|---|---|
| numerator | Signed integer numerator | string (integer/arbitrary); maximum text length 200; pattern ^-?(0|[1-9][0-9]*)$ | Required |
| denominator | Signed integer denominator | string (integer/arbitrary); maximum text length 200; pattern ^-?(0|[1-9][0-9]*)$ | Required |
Parse both signed decimal strings with BigInt. Handle zero denominator, adjust for a negative denominator, remove powers of two using residue mod eight, then evaluate the odd part.
(a/0)=1 if |a|=1, otherwise 0; combine denominator sign, factors of two and odd-part Jacobi symbol The symbols and boundary conditions refer to the supplied fields and the implementation described above.
| Output field | Meaning | Value type |
|---|---|---|
| symbol | Kronecker symbol | integer |
{
"input": {
"numerator": "3",
"denominator": "8"
},
"output": {
"symbol": -1
}
}Limitations and assumptions
A symbol of one need not prove quadratic residuosity for a composite denominator. This returns only −1, 0 or 1 and accepts at most 200 characters per input.
Use the declared integer bounds and string formats. Large exact outputs remain decimal strings; numerical approximations use ordinary floating-point arithmetic. A rejected input is different from a valid zero count or false membership flag.
Alternative or Complementary methods
Choose Jacobi Symbol when you need its operation: Return (a/n) for a signed decimal numerator and positive odd decimal denominator, including (a/1)=1; a value of one does not prove a quadratic residue for composite n. Choose Legendre Symbol for the complementary operation: Return the Legendre symbol (a/p) for a signed decimal integer and an odd prime p, using Euler’s criterion and trial primality testing limited to p≤10¹².
Background reference: Arithmetic functions: kronecker symbol.
References
-
Arithmetic functions: kronecker symbol — SageMath developers
Similar or alternative tools
- Jacobi Symbol
Return (a/n) for a signed decimal numerator and positive odd decimal denominator, including (a/1)=1; a value of one does not prove a quadratic residue for composite n.
- Legendre Symbol
Return the Legendre symbol (a/p) for a signed decimal integer and an odd prime p, using Euler’s criterion and trial primality testing limited to p≤10¹².