## Class BigIntegerMath

• ```@GwtCompatible(emulated=true)
public final class BigIntegerMath
extends Object```
A class for arithmetic on values of type `BigInteger`.

The implementations of many methods in this class are based on material from Henry S. Warren, Jr.'s Hacker's Delight, (Addison Wesley, 2002).

Similar functionality for `int` and for `long` can be found in `IntMath` and `LongMath` respectively.

Since:
11.0
Author:
Louis Wasserman
• ### Method Summary

All Methods
Modifier and Type Method Description
`static BigInteger` ```binomial​(int n, int k)```
Returns `n` choose `k`, also known as the binomial coefficient of `n` and `k`, that is, `n! / (k! (n - k)!)`.
`static BigInteger` `ceilingPowerOfTwo​(BigInteger x)`
Returns the smallest power of two greater than or equal to `x`.
`static BigInteger` ```divide​(BigInteger p, BigInteger q, RoundingMode mode)```
Returns the result of dividing `p` by `q`, rounding using the specified ``` RoundingMode```.
`static BigInteger` `factorial​(int n)`
Returns `n!`, that is, the product of the first `n` positive integers, or `1` if `n == 0`.
`static BigInteger` `floorPowerOfTwo​(BigInteger x)`
Returns the largest power of two less than or equal to `x`.
`static boolean` `isPowerOfTwo​(BigInteger x)`
Returns `true` if `x` represents a power of two.
`static int` ```log10​(BigInteger x, RoundingMode mode)```
Returns the base-10 logarithm of `x`, rounded according to the specified rounding mode.
`static int` ```log2​(BigInteger x, RoundingMode mode)```
Returns the base-2 logarithm of `x`, rounded according to the specified rounding mode.
`static double` ```roundToDouble​(BigInteger x, RoundingMode mode)```
Returns `x`, rounded to a `double` with the specified rounding mode.
`static BigInteger` ```sqrt​(BigInteger x, RoundingMode mode)```
Returns the square root of `x`, rounded with the specified rounding mode.
• ### Methods inherited from class java.lang.Object

`clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait`
• ### Method Detail

• #### ceilingPowerOfTwo

```@Beta
public static BigInteger ceilingPowerOfTwo​(BigInteger x)```
Returns the smallest power of two greater than or equal to `x`. This is equivalent to `BigInteger.valueOf(2).pow(log2(x, CEILING))`.
Throws:
`IllegalArgumentException` - if `x <= 0`
Since:
20.0
• #### floorPowerOfTwo

```@Beta
public static BigInteger floorPowerOfTwo​(BigInteger x)```
Returns the largest power of two less than or equal to `x`. This is equivalent to ``` BigInteger.valueOf(2).pow(log2(x, FLOOR))```.
Throws:
`IllegalArgumentException` - if `x <= 0`
Since:
20.0
• #### isPowerOfTwo

`public static boolean isPowerOfTwo​(BigInteger x)`
Returns `true` if `x` represents a power of two.
• #### log2

```public static int log2​(BigInteger x,
RoundingMode mode)```
Returns the base-2 logarithm of `x`, rounded according to the specified rounding mode.
Throws:
`IllegalArgumentException` - if `x <= 0`
`ArithmeticException` - if `mode` is `RoundingMode.UNNECESSARY` and `x` is not a power of two
• #### log10

```@GwtIncompatible
public static int log10​(BigInteger x,
RoundingMode mode)```
Returns the base-10 logarithm of `x`, rounded according to the specified rounding mode.
Throws:
`IllegalArgumentException` - if `x <= 0`
`ArithmeticException` - if `mode` is `RoundingMode.UNNECESSARY` and `x` is not a power of ten
• #### sqrt

```@GwtIncompatible
public static BigInteger sqrt​(BigInteger x,
RoundingMode mode)```
Returns the square root of `x`, rounded with the specified rounding mode.
Throws:
`IllegalArgumentException` - if `x < 0`
`ArithmeticException` - if `mode` is `RoundingMode.UNNECESSARY` and ``` sqrt(x)``` is not an integer
• #### roundToDouble

```@GwtIncompatible
public static double roundToDouble​(BigInteger x,
RoundingMode mode)```
Returns `x`, rounded to a `double` with the specified rounding mode. If `x` is precisely representable as a `double`, its `double` value will be returned; otherwise, the rounding will choose between the two nearest representable values with ``` mode```.

For the case of `RoundingMode.HALF_DOWN`, `HALF_UP`, and `HALF_EVEN`, infinite `double` values are considered infinitely far away. For example, 2^2000 is not representable as a double, but `roundToDouble(BigInteger.valueOf(2).pow(2000), HALF_UP)` will return `Double.MAX_VALUE`, not `Double.POSITIVE_INFINITY`.

For the case of `RoundingMode.HALF_EVEN`, this implementation uses the IEEE 754 default rounding mode: if the two nearest representable values are equally near, the one with the least significant bit zero is chosen. (In such cases, both of the nearest representable values are even integers; this method returns the one that is a multiple of a greater power of two.)

Throws:
`ArithmeticException` - if `mode` is `RoundingMode.UNNECESSARY` and `x` is not precisely representable as a `double`
Since:
30.0
• #### divide

```@GwtIncompatible
public static BigInteger divide​(BigInteger p,
BigInteger q,
RoundingMode mode)```
Returns the result of dividing `p` by `q`, rounding using the specified ``` RoundingMode```.
Throws:
`ArithmeticException` - if `q == 0`, or if `mode == UNNECESSARY` and `a` is not an integer multiple of `b`
• #### factorial

`public static BigInteger factorial​(int n)`
Returns `n!`, that is, the product of the first `n` positive integers, or `1` if `n == 0`.

Warning: the result takes O(n log n) space, so use cautiously.

This uses an efficient binary recursive algorithm to compute the factorial with balanced multiplies. It also removes all the 2s from the intermediate products (shifting them back in at the end).

Throws:
`IllegalArgumentException` - if `n < 0`
• #### binomial

```public static BigInteger binomial​(int n,
int k)```
Returns `n` choose `k`, also known as the binomial coefficient of `n` and `k`, that is, `n! / (k! (n - k)!)`.

Warning: the result can take as much as O(k log n) space.

Throws:
`IllegalArgumentException` - if `n < 0`, `k < 0`, or `k > n`