MathPHP\NumberTheory\Integer Class Reference

Static Public Member Functions
static	isPerfectNumber (int $n)
	Detect if an integer is a perfect number. More...
static	isPerfectPower (int $n)
	Detect if an integer is a perfect power. More...
static	perfectPower (int $n)
	If n is a perfect power, compute an m and k such that mᵏ = n. More...
static	primeFactorization (int $n)
	Prime factorization The prime factors of an integer. More...
static	coprime (int $a, int $b)
	Coprime (relatively prime, mututally prime) Two integers a and b are said to be coprime if the only positive integer that divides both of them is 1. More...
static	isOdd (int $x)
	Odd number. More...
static	isEven (int $x)
	Even number. More...

Member Function Documentation

coprime()

static MathPHP\NumberTheory\Integer::coprime ( int  $a, int  $b  )

static

Coprime (relatively prime, mututally prime) Two integers a and b are said to be coprime if the only positive integer that divides both of them is 1.

That is, the only common positive factor of the two numbers is 1. This is equivalent to their greatest common divisor being 1. https://en.wikipedia.org/wiki/Coprime_integers

Parameters

int	$a
int	$b

Returns

bool true if a and b are coprime; false otherwise

isEven()

static MathPHP\NumberTheory\Integer::isEven ( int  $x )

static

Even number.

Parameters

int

$x

Returns

bool true if x is even; false otherwise

isOdd()

static MathPHP\NumberTheory\Integer::isOdd ( int  $x )

static

Odd number.

Parameters

int

$x

Returns

bool true if x is odd; false otherwise

isPerfectNumber()

static MathPHP\NumberTheory\Integer::isPerfectNumber ( int  $n )

static

Detect if an integer is a perfect number.

A perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself

See also

https://en.wikipedia.org/wiki/Perfect_number

Parameters

int

$n

Returns

bool

isPerfectPower()

static MathPHP\NumberTheory\Integer::isPerfectPower ( int  $n )

static

Detect if an integer is a perfect power.

A perfect power is a positive integer that can be expressed as an integer power of another positive integer. If n is a perfect power, then exists m > 1 and k > 1 such that mᵏ = n. https://en.wikipedia.org/wiki/Perfect_power

Algorithm: For each divisor of n (as m), consider all possible values of k from 2 to log₂n.

• If mᵏ = n, return true
• If exhaust all possible mᵏ combinations, return false.

Parameters

int

$n

Returns

bool True if n is a perfect power; false otherwise.

perfectPower()

static MathPHP\NumberTheory\Integer::perfectPower ( int  $n )

static

If n is a perfect power, compute an m and k such that mᵏ = n.

A perfect power is a positive integer that can be expressed as an integer power of another positive integer. If n is a perfect power, then exists m > 1 and k > 1 such that mᵏ = n. https://en.wikipedia.org/wiki/Perfect_power

Algorithm: For each divisor of n (as m), consider all possible values of k from 2 to log₂n.

• If mᵏ = n, return m and k
• If exhaust all possible mᵏ combinations, return empty array.

An integer n could have multiple perfect power scenarios. Only one is returned.

Parameters

int

$n

Returns

array [m, k]

primeFactorization()

static MathPHP\NumberTheory\Integer::primeFactorization ( int  $n )

static

Prime factorization The prime factors of an integer.

https://en.wikipedia.org/wiki/Prime_factor

Todo:

Use a better algorithm.

Parameters

int

$n

Returns

array of prime factors

Exceptions

Exception

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The documentation for this class was generated from the following file:

• src/NumberTheory/Integer.php