Quadratic Residue Calculator

Determine if a number is a quadratic residue modulo p and find its square roots.

Enter Values

Number a

Modulus p

Definition

An integer a is a quadratic residue modulo n if:

x^2 ≡ a (mod n)

has a solution for some integer x.

Euler's Criterion

For odd prime p and gcd(a, p) = 1:

a^((p-1)/2) ≡ 1 (mod p) iff a is QR

Is 3 a Quadratic Residue mod 13?

YES

Legendre symbol: (3/13) = 1

Modulus Type

Prime

# QR (excl. 0)

Square Roots

Solutions to x^2 ≡ 3 (mod 13):

x = 4x = 9

Quadratic Residues mod 13

13491012

Non-Residues mod 13

2567811

What Is a Quadratic Residue Calculator?

A quadratic residue calculator helps you perform quadratic residue 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 Quadratic Residue Formula

The calculator applies the following mathematical relationships:

Quadratic Residue 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

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

Real-World Applications

Quadratic Residue 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 quadratic residue calculations.

Worked Examples

Example Calculation

Problem:

Use the default values to compute the result.

Solution Steps:

1Enter a = 3.
2Enter p = 13.
3The calculator computes the result using the appropriate formula.
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 quadratic residue 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.

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.