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Legendre Symbol Calculator

Calculate the Legendre symbol (a/p) for an integer a and odd prime p.

Enter Values

Definition

The Legendre symbol (a/p) is defined as:

(a/p) = 0 if p divides a

(a/p) = 1 if a is a QR mod p

(a/p) = -1 if a is a NR mod p

Euler's Criterion

(a/p) ≡ a^((p-1)/2) (mod p)

Properties

  • (ab/p) = (a/p)(b/p)
  • (a^2/p) = 1
  • (-1/p) = (-1)^((p-1)/2)
  • (2/p) = (-1)^((p^2-1)/8)

Legendre Symbol

(7/13) = -1

a mod p
7
a^((p-1)/2) mod p
12

Interpretation

7 is a non-residue mod 13

Legendre Symbols for 1 to 12

123456789101112
QR: 6NR: 6

Calculation Method

Euler's Criterion

7^6 mod 13 = 12 ≡ -1

What Is a Legendre Symbol Calculator?

A legendre symbol calculator helps you perform legendre symbol calculations quickly and accurately. Enter your values and get instant results with step-by-step breakdowns showing exactly how each result was derived.

This calculator handles 2 input values: a, p. Results are computed using standard mathematical formulas and displayed with precision suitable for homework, professional work, and quick references.

The Legendre Symbol Formula

The calculator applies the following mathematical relationships:

Legendre Symbol Formula

Computed from input values using standard formulas

Where:

  • Input= Enter values in the input fields to compute results

Understanding the Results

The results display shows the computed value{s} along with related quantities. Each result is computed using JavaScript's built-in Math functions (Math.PI, Math.sqrt, etc.) for maximum precision.

All results are shown to four decimal places by default, which is sufficient for most practical applications including construction, engineering, and academic work.

How to Use This Calculator

  1. Enter a: Type a value in the a field. Default value is 7.
  2. Enter p: Type a value in the p field. Default value is 13.
  3. Read the results: The calculator updates immediately as you type, showing computed values with full step-by-step breakdowns.

Real-World Applications

Legendre Symbol calculations appear in numerous fields. In education, students use them to verify homework answers and understand the underlying formulas. In engineering, these calculations inform design decisions and safety margins. In everyday life, quick calculations help with home improvement projects, budgeting, and planning.

The specific formulas used by this calculator are standard in the field and can be verified in any mathematics or engineering textbook. Bookmark this page as a quick reference whenever you need to perform legendre symbol calculations.

Worked Examples

Example Calculation

Problem:

Use the default values to compute the result.

Solution Steps:

  1. 1Enter a = 7.
  2. 2Enter p = 13.
  3. 3The calculator computes the result using the appropriate formula.
  4. 4Review the step-by-step breakdown to understand the process.

Result:

The computed result is displayed in the highlighted result card above.

Tips & Best Practices

  • ✓Double-check your inputs — a single typo can produce dramatically different results.
  • ✓Use consistent units throughout — don't mix centimeters with inches or meters with feet.
  • ✓Review the step-by-step breakdown to verify that the formula was applied correctly for your inputs.
  • ✓Bookmark this page for quick access to legendre symbol calculations whenever needed.
  • ✓For very large or small numbers, the calculator may display results in exponential notation.
  • ✓Compare results with manual calculations occasionally to build confidence in the tool and your math skills.

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Frequently Asked Questions

Calculations use JavaScript's built-in Math functions with double-precision floating-point arithmetic, providing approximately 15 significant digits of accuracy. Displayed values are rounded to four decimal places for readability while maintaining sufficient precision for most applications.
This is a web-based calculator that requires an internet connection to load. Once loaded, all calculations are performed client-side in your browser, so no data is sent to any server.
The calculator is unit-agnostic — enter your values in any consistent unit system (metric or imperial), and results will be in the corresponding units. Mixing unit systems will produce incorrect results.
Yes, this calculator is completely free with no registration required. Use it as many times as you need for homework, professional work, or personal projects.
If you find an issue or have a suggestion for improving this calculator, please reach out through the contact options on the site. We're always looking to make our tools more useful and accurate.

Sources & References

Last updated: 2026-06-06

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Editorial Note

MyCalcBuddy Editorial Team

This page is maintained as an educational calculator reference.

Source

Formula Source: Handbook of Mathematical Functions

by Abramowitz & Stegun

UpdatedLast reviewed: May 2026
CheckedFormula checks are based on standard references and internal QA review.