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Quadratic Residue Calculator

Work out quadratic residue instantly with clear inputs, formula shown and shareable results.

About Quadratic Residue Calculator

A quadratic residue modulo m is a value congruent to some perfect square. Unlike ordinary square roots, a modular square root may have zero, one, two or more solutions depending on the factorisation of m.

Formula

Quadratic residue

a is a residue mod m if x² ≡ a (mod m) has a solution

Worked example

With the default values below, Quadratic Residue Calculator returns Result of 9 is a quadratic residue mod 23. Change any input above and every figure here is recalculated in your browser.

Inputs used

Value a
9
Modulus mminimum 2
23

Results

Result
9 is a quadratic residue mod 23
Modular square roots
3, 20
How many roots
2
Distinct quadratic residues mod m
12

Step by step

  1. 1Values used
  2. 2Quadratic residue
  3. 3Result= 9 is a quadratic residue mod 23
  4. 4Modular square roots= 3, 20
  5. 5How many roots= 2
  6. 6Distinct quadratic residues mod m= 12

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Is 9 a residue mod 23?

Yes, with roots 3 and 20, since 20 ≡ −3 and (−3)² = 9.

Why can there be more than two roots?

For composite moduli the Chinese remainder theorem multiplies the root counts of each prime power factor.

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