# CompositeQ

CompositeQ[n]

yields True if n is a composite number, and yields False otherwise.

# Details and Options • CompositeQ is typically used to test whether an integer is a composite number.
• A composite number is a positive number that is the product of two integers other than 1.
• • CompositeQ[n] returns False unless n is manifestly a composite number.
• For negative integer n, CompositeQ[n] is effectively equivalent to CompositeQ[-n].
• With the setting , CompositeQ determines whether a number is a composite number over Gaussian integers.
• CompositeQ[m+In] automatically works over Gaussian integers.

# Examples

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## Basic Examples(2)

Test whether a number is composite:

The number is not composite:

## Scope(4)

CompositeQ works over integers:

Gaussian integers:

Test for large integers:

## Options(1)

### GaussianIntegers(1)

Test whether is composite over integers:

Gaussian integers:

## Applications(9)

### Basic Applications(3)

Highlight composite numbers:

Generate the  composite number:

Generate random composite numbers:

The distribution of Gaussian composite numbers:

### Number Theory(6)

Recognize Sierpiński numbers k where is always composite:

Recognize powerful numbers n whose prime factors are all repeated:

All perfect powers are powerful numbers:

Recognize base b pseudoprimes, composite numbers n such that :

Find all base pseudoprimes below :

Find all base pseudoprimes below :

Find large composite numbers of the form :

The distribution of composite numbers over integers:

Plot the distribution:

The distribution of composite numbers over the Gaussian integers:

Plot the distribution:

## Properties & Relations(6)

Primes represents the domain of all prime numbers:

PrimeQ gives False for all composite numbers:

CompositeQ gives False for all primes:

Composite numbers cannot be a MersennePrimeExponent:

Composite numbers have at least two prime factors including multiplicities:

Composite numbers that are the  power of a prime number have exactly divisors:

## Neat Examples(2)

Plot composite numbers that are the sum of three squares:

Plot the Ulam spiral where numbers are colored based on their compositeness: