Module java.base
Package java.math

Class BigInteger

All Implemented Interfaces:
Serializable, Comparable<BigInteger>

public class BigInteger extends Number implements Comparable<BigInteger>
Immutable arbitrary-precision integers. All operations behave as if BigIntegers were represented in two's-complement notation (like Java's primitive integer types). BigInteger provides analogues to all of Java's primitive integer operators, and all relevant methods from java.lang.Math. Additionally, BigInteger provides operations for modular arithmetic, GCD calculation, primality testing, prime generation, bit manipulation, and a few other miscellaneous operations.

Semantics of arithmetic operations exactly mimic those of Java's integer arithmetic operators, as defined in The Java Language Specification. For example, division by zero throws an ArithmeticException, and division of a negative by a positive yields a negative (or zero) remainder.

Semantics of shift operations extend those of Java's shift operators to allow for negative shift distances. A right-shift with a negative shift distance results in a left shift, and vice-versa. The unsigned right shift operator (>>>) is omitted since this operation only makes sense for a fixed sized word and not for a representation conceptually having an infinite number of leading virtual sign bits.

Semantics of bitwise logical operations exactly mimic those of Java's bitwise integer operators. The binary operators (and, or, xor) implicitly perform sign extension on the shorter of the two operands prior to performing the operation.

Comparison operations perform signed integer comparisons, analogous to those performed by Java's relational and equality operators.

Modular arithmetic operations are provided to compute residues, perform exponentiation, and compute multiplicative inverses. These methods always return a non-negative result, between 0 and (modulus - 1), inclusive.

Bit operations operate on a single bit of the two's-complement representation of their operand. If necessary, the operand is sign-extended so that it contains the designated bit. None of the single-bit operations can produce a BigInteger with a different sign from the BigInteger being operated on, as they affect only a single bit, and the arbitrarily large abstraction provided by this class ensures that conceptually there are infinitely many "virtual sign bits" preceding each BigInteger.

For the sake of brevity and clarity, pseudo-code is used throughout the descriptions of BigInteger methods. The pseudo-code expression (i + j) is shorthand for "a BigInteger whose value is that of the BigInteger i plus that of the BigInteger j." The pseudo-code expression (i == j) is shorthand for "true if and only if the BigInteger i represents the same value as the BigInteger j." Other pseudo-code expressions are interpreted similarly.

All methods and constructors in this class throw NullPointerException when passed a null object reference for any input parameter. BigInteger must support values in the range -2Integer.MAX_VALUE (exclusive) to +2Integer.MAX_VALUE (exclusive) and may support values outside of that range. An ArithmeticException is thrown when a BigInteger constructor or method would generate a value outside of the supported range. The range of probable prime values is limited and may be less than the full supported positive range of BigInteger. The range must be at least 1 to 2500000000.

API Note:
As BigInteger values are arbitrary precision integers, the algorithmic complexity of the methods of this class varies and may be superlinear in the size of the input. For example, a method like intValue() would be expected to run in O(1), that is constant time, since with the current internal representation only a fixed-size component of the BigInteger needs to be accessed to perform the conversion to int. In contrast, a method like not() would be expected to run in O(n) time where n is the size of the BigInteger in bits, that is, to run in time proportional to the size of the input. For multiplying two BigInteger values of size n, a naive multiplication algorithm would run in time O(n2) and theoretical results indicate a multiplication algorithm for numbers using this category of representation must run in at least O(n log n). Common multiplication algorithms between the bounds of the naive and theoretical cases include the Karatsuba multiplication (O(n1.585)) and 3-way Toom-Cook multiplication (O(n1.465)).

A particular implementation of multiply is free to switch between different algorithms for different inputs, such as to improve actual running time to produce the product by using simpler algorithms for smaller inputs even if the simpler algorithm has a larger asymptotic complexity.

Operations may also allocate and compute on intermediate results, potentially those allocations may be as large as in proportion to the running time of the algorithm.

Users of BigInteger concerned with bounding the running time or space of operations can screen out BigInteger values above a chosen magnitude.

Implementation Note:
In the reference implementation, BigInteger constructors and operations throw ArithmeticException when the result is out of the supported range of -2Integer.MAX_VALUE (exclusive) to +2Integer.MAX_VALUE (exclusive).
See Java Language Specification:
4.2.2 Integer Operations
See Also: