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.

Description

The Jacobi symbol extends Legendre’s prime-modulus symbol to a positive odd denominator.

The Jacobi symbol extends Legendre’s prime-modulus symbol to a positive odd denominator. For composite denominators, a value of one is necessary in some residue settings but does not prove a square residue.^[1]

The tool returns symbol. The indexing and output conventions below determine how to read these values.

When to use Jacobi Symbol

Use this for the following task: 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. Use the input and output conventions below to reproduce the intended mathematical quantity.

How to use Jacobi 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	Positive odd denominator	string (integer/arbitrary); maximum text length 200; pattern ^[1-9][0-9]*$	Required

Normalize the signed BigInt numerator, remove powers of two with mod-eight sign changes, swap numerator and denominator with reciprocity signs, and reduce until termination.

(a/n) from removal of powers of two and quadratic reciprocity; (a/1)=1

The symbols and boundary conditions refer to the supplied fields and the implementation described above.

Output field	Meaning	Value type
symbol	Jacobi symbol	integer

json

{
  "input": {
    "numerator": "5",
    "denominator": "21"
  },
  "output": {
    "symbol": 1
  }
}

Limitations and assumptions

The denominator must be positive and odd. Each input is capped at 200 characters; use Kronecker for even, negative or zero denominators.

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 Legendre Symbol when you need its 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. Choose Kronecker Symbol for the complementary operation: Extend the Jacobi symbol to signed decimal denominators, including factors of two, negative denominators, and (a/0)=1 exactly when |a|=1.

Background reference: Arithmetic functions: jacobi symbol.

References

Arithmetic functions: jacobi symbol — SageMath developers

Similar or alternative tools

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.
Kronecker Symbol
Extend the Jacobi symbol to signed decimal denominators, including factors of two, negative denominators, and (a/0)=1 exactly when |a|=1.