Legendre Symbol Calculator

Work out legendre symbol instantly with clear inputs, formula shown and shareable results.

About Legendre Symbol Calculator

The Legendre symbol (a|p) is 1 when a is a non-zero square modulo p, −1 when it is not, and 0 when p divides a. Euler's criterion computes it as a^((p−1)/2) mod p, reading p − 1 as −1.

Formula

Euler's criterion

(a|p) ≡ a^((p−1)/2) (mod p)

Worked example

With the default values below, Legendre Symbol Calculator returns Legendre symbol (a|p) of 1. Change any input above and every figure here is recalculated in your browser.

Inputs used

Numerator a: 5
Odd prime pminimum 3: 11

Results

Legendre symbol (a|p): 1
Meaning: 5 is a quadratic residue mod 11
a^((p−1)/2) mod p: 1
Number of non-zero quadratic residues mod p: 5

Step by step

1Values used
2Euler's criterion
3Legendre symbol (a|p)= 1
4Meaning= 5 is a quadratic residue mod 11
5a^((p−1)/2) mod p= 1
6Number of non-zero quadratic residues mod p= 5

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 5 a square mod 11?

5⁵ mod 11 = 1, so yes: 4² = 16 ≡ 5 (mod 11).

How many residues are there?

Exactly (p − 1)/2 non-zero ones, since x and −x always square to the same value.