
Java™ Platform Standard Ed. 6 

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java.lang.Object java.lang.Number java.math.BigInteger
public class BigInteger
Immutable arbitraryprecision integers. All operations behave as if BigIntegers were represented in two'scomplement 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.
All of the details in the Spec concerning overflow are ignored, as
BigIntegers are made as large as necessary to accommodate the results of an
operation.
Semantics of shift operations extend those of Java's shift operators
to allow for negative shift distances. A rightshift with a negative
shift distance results in a left shift, and viceversa. The unsigned
right shift operator (>>>
) is omitted, as this operation makes
little sense in combination with the "infinite word size" abstraction
provided by this class.
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 nonnegative result, between 0
and (modulus  1)
,
inclusive.
Bit operations operate on a single bit of the two'scomplement representation of their operand. If necessary, the operand is sign extended so that it contains the designated bit. None of the singlebit operations can produce a BigInteger with a different sign from the BigInteger being operated on, as they affect only a single bit, and the "infinite word size" abstraction provided by this class ensures that there are infinitely many "virtual sign bits" preceding each BigInteger.
For the sake of brevity and clarity, pseudocode is used throughout the
descriptions of BigInteger methods. The pseudocode expression
(i + j)
is shorthand for "a BigInteger whose value is
that of the BigInteger i
plus that of the BigInteger j
."
The pseudocode expression (i == j)
is shorthand for
"true
if and only if the BigInteger i
represents the same
value as the BigInteger j
." Other pseudocode expressions are
interpreted similarly.
All methods and constructors in this class throw
NullPointerException
when passed
a null object reference for any input parameter.
BigDecimal
,
Serialized FormField Summary  

static BigInteger 
ONE
The BigInteger constant one. 
static BigInteger 
TEN
The BigInteger constant ten. 
static BigInteger 
ZERO
The BigInteger constant zero. 
Constructor Summary  

BigInteger(byte[] val)
Translates a byte array containing the two'scomplement binary representation of a BigInteger into a BigInteger. 

BigInteger(int signum,
byte[] magnitude)
Translates the signmagnitude representation of a BigInteger into a BigInteger. 

BigInteger(int bitLength,
int certainty,
Random rnd)
Constructs a randomly generated positive BigInteger that is probably prime, with the specified bitLength. 

BigInteger(int numBits,
Random rnd)
Constructs a randomly generated BigInteger, uniformly distributed over the range 0 to (2^{numBits}  1), inclusive. 

BigInteger(String val)
Translates the decimal String representation of a BigInteger into a BigInteger. 

BigInteger(String val,
int radix)
Translates the String representation of a BigInteger in the specified radix into a BigInteger. 
Method Summary  

BigInteger 
abs()
Returns a BigInteger whose value is the absolute value of this BigInteger. 
BigInteger 
add(BigInteger val)
Returns a BigInteger whose value is (this + val) . 
BigInteger 
and(BigInteger val)
Returns a BigInteger whose value is (this & val) . 
BigInteger 
andNot(BigInteger val)
Returns a BigInteger whose value is (this & ~val) . 
int 
bitCount()
Returns the number of bits in the two's complement representation of this BigInteger that differ from its sign bit. 
int 
bitLength()
Returns the number of bits in the minimal two'scomplement representation of this BigInteger, excluding a sign bit. 
BigInteger 
clearBit(int n)
Returns a BigInteger whose value is equivalent to this BigInteger with the designated bit cleared. 
int 
compareTo(BigInteger val)
Compares this BigInteger with the specified BigInteger. 
BigInteger 
divide(BigInteger val)
Returns a BigInteger whose value is (this / val) . 
BigInteger[] 
divideAndRemainder(BigInteger val)
Returns an array of two BigIntegers containing (this / val)
followed by (this % val) . 
double 
doubleValue()
Converts this BigInteger to a double . 
boolean 
equals(Object x)
Compares this BigInteger with the specified Object for equality. 
BigInteger 
flipBit(int n)
Returns a BigInteger whose value is equivalent to this BigInteger with the designated bit flipped. 
float 
floatValue()
Converts this BigInteger to a float . 
BigInteger 
gcd(BigInteger val)
Returns a BigInteger whose value is the greatest common divisor of abs(this) and abs(val) . 
int 
getLowestSetBit()
Returns the index of the rightmost (lowestorder) one bit in this BigInteger (the number of zero bits to the right of the rightmost one bit). 
int 
hashCode()
Returns the hash code for this BigInteger. 
int 
intValue()
Converts this BigInteger to an int . 
boolean 
isProbablePrime(int certainty)
Returns true if this BigInteger is probably prime,
false if it's definitely composite. 
long 
longValue()
Converts this BigInteger to a long . 
BigInteger 
max(BigInteger val)
Returns the maximum of this BigInteger and val . 
BigInteger 
min(BigInteger val)
Returns the minimum of this BigInteger and val . 
BigInteger 
mod(BigInteger m)
Returns a BigInteger whose value is (this mod m ). 
BigInteger 
modInverse(BigInteger m)
Returns a BigInteger whose value is (this ^{1} mod m) . 
BigInteger 
modPow(BigInteger exponent,
BigInteger m)
Returns a BigInteger whose value is (this^{exponent} mod m). 
BigInteger 
multiply(BigInteger val)
Returns a BigInteger whose value is (this * val) . 
BigInteger 
negate()
Returns a BigInteger whose value is (this) . 
BigInteger 
nextProbablePrime()
Returns the first integer greater than this BigInteger that
is probably prime. 
BigInteger 
not()
Returns a BigInteger whose value is (~this) . 
BigInteger 
or(BigInteger val)
Returns a BigInteger whose value is (this  val) . 
BigInteger 
pow(int exponent)
Returns a BigInteger whose value is (this^{exponent}). 
static BigInteger 
probablePrime(int bitLength,
Random rnd)
Returns a positive BigInteger that is probably prime, with the specified bitLength. 
BigInteger 
remainder(BigInteger val)
Returns a BigInteger whose value is (this % val) . 
BigInteger 
setBit(int n)
Returns a BigInteger whose value is equivalent to this BigInteger with the designated bit set. 
BigInteger 
shiftLeft(int n)
Returns a BigInteger whose value is (this << n) . 
BigInteger 
shiftRight(int n)
Returns a BigInteger whose value is (this >> n) . 
int 
signum()
Returns the signum function of this BigInteger. 
BigInteger 
subtract(BigInteger val)
Returns a BigInteger whose value is (this  val) . 
boolean 
testBit(int n)
Returns true if and only if the designated bit is set. 
byte[] 
toByteArray()
Returns a byte array containing the two'scomplement representation of this BigInteger. 
String 
toString()
Returns the decimal String representation of this BigInteger. 
String 
toString(int radix)
Returns the String representation of this BigInteger in the given radix. 
static BigInteger 
valueOf(long val)
Returns a BigInteger whose value is equal to that of the specified long . 
BigInteger 
xor(BigInteger val)
Returns a BigInteger whose value is (this ^ val) . 
Methods inherited from class java.lang.Number 

byteValue, shortValue 
Methods inherited from class java.lang.Object 

clone, finalize, getClass, notify, notifyAll, wait, wait, wait 
Field Detail 

public static final BigInteger ZERO
public static final BigInteger ONE
public static final BigInteger TEN
Constructor Detail 

public BigInteger(byte[] val)
val
 bigendian two'scomplement binary representation of
BigInteger.
NumberFormatException
 val
is zero bytes long.public BigInteger(int signum, byte[] magnitude)
signum
 signum of the number (1 for negative, 0 for zero, 1
for positive).magnitude
 bigendian binary representation of the magnitude of
the number.
NumberFormatException
 signum
is not one of the three
legal values (1, 0, and 1), or signum
is 0 and
magnitude
contains one or more nonzero bytes.public BigInteger(String val, int radix)
Character.digit
. The String may not contain any extraneous
characters (whitespace, for example).
val
 String representation of BigInteger.radix
 radix to be used in interpreting val
.
NumberFormatException
 val
is not a valid representation
of a BigInteger in the specified radix, or radix
is
outside the range from Character.MIN_RADIX
to
Character.MAX_RADIX
, inclusive.Character.digit(char, int)
public BigInteger(String val)
Character.digit
.
The String may not contain any extraneous characters (whitespace, for
example).
val
 decimal String representation of BigInteger.
NumberFormatException
 val
is not a valid representation
of a BigInteger.Character.digit(char, int)
public BigInteger(int numBits, Random rnd)
0
to (2^{numBits}  1), inclusive.
The uniformity of the distribution assumes that a fair source of random
bits is provided in rnd
. Note that this constructor always
constructs a nonnegative BigInteger.
numBits
 maximum bitLength of the new BigInteger.rnd
 source of randomness to be used in computing the new
BigInteger.
IllegalArgumentException
 numBits
is negative.bitLength()
public BigInteger(int bitLength, int certainty, Random rnd)
It is recommended that the probablePrime
method be used in preference to this constructor unless there
is a compelling need to specify a certainty.
bitLength
 bitLength of the returned BigInteger.certainty
 a measure of the uncertainty that the caller is
willing to tolerate. The probability that the new BigInteger
represents a prime number will exceed
(1  1/2^{certainty}). The execution time of
this constructor is proportional to the value of this parameter.rnd
 source of random bits used to select candidates to be
tested for primality.
ArithmeticException
 bitLength < 2
.bitLength()
Method Detail 

public static BigInteger probablePrime(int bitLength, Random rnd)
bitLength
 bitLength of the returned BigInteger.rnd
 source of random bits used to select candidates to be
tested for primality.
bitLength
bits that is probably prime
ArithmeticException
 bitLength < 2
.bitLength()
public BigInteger nextProbablePrime()
BigInteger
that
is probably prime. The probability that the number returned by this
method is composite does not exceed 2^{100}. This method will
never skip over a prime when searching: if it returns p
, there
is no prime q
such that this < q < p
.
BigInteger
that
is probably prime.
ArithmeticException
 this < 0
.public static BigInteger valueOf(long val)
long
. This "static factory method" is
provided in preference to a (long
) constructor
because it allows for reuse of frequently used BigIntegers.
val
 value of the BigInteger to return.
public BigInteger add(BigInteger val)
(this + val)
.
val
 value to be added to this BigInteger.
this + val
public BigInteger subtract(BigInteger val)
(this  val)
.
val
 value to be subtracted from this BigInteger.
this  val
public BigInteger multiply(BigInteger val)
(this * val)
.
val
 value to be multiplied by this BigInteger.
this * val
public BigInteger divide(BigInteger val)
(this / val)
.
val
 value by which this BigInteger is to be divided.
this / val
ArithmeticException
 val==0
public BigInteger[] divideAndRemainder(BigInteger val)
(this / val)
followed by (this % val)
.
val
 value by which this BigInteger is to be divided, and the
remainder computed.
(this / val)
is the initial element, and the remainder (this % val)
is the final element.
ArithmeticException
 val==0
public BigInteger remainder(BigInteger val)
(this % val)
.
val
 value by which this BigInteger is to be divided, and the
remainder computed.
this % val
ArithmeticException
 val==0
public BigInteger pow(int exponent)
exponent
is an integer rather than a BigInteger.
exponent
 exponent to which this BigInteger is to be raised.
ArithmeticException
 exponent
is negative. (This would
cause the operation to yield a noninteger value.)public BigInteger gcd(BigInteger val)
abs(this)
and abs(val)
. Returns 0 if
this==0 && val==0
.
val
 value with which the GCD is to be computed.
GCD(abs(this), abs(val))
public BigInteger abs()
abs(this)
public BigInteger negate()
(this)
.
this
public int signum()
public BigInteger mod(BigInteger m)
(this mod m
). This method
differs from remainder
in that it always returns a
nonnegative BigInteger.
m
 the modulus.
this mod m
ArithmeticException
 m <= 0
remainder(java.math.BigInteger)
public BigInteger modPow(BigInteger exponent, BigInteger m)
pow
, this
method permits negative exponents.)
exponent
 the exponent.m
 the modulus.
ArithmeticException
 m <= 0
modInverse(java.math.BigInteger)
public BigInteger modInverse(BigInteger m)
(this
^{1} mod m)
.
m
 the modulus.
this
^{1} mod m
.
ArithmeticException
 m <= 0
, or this BigInteger
has no multiplicative inverse mod m (that is, this BigInteger
is not relatively prime to m).public BigInteger shiftLeft(int n)
(this << n)
.
The shift distance, n
, may be negative, in which case
this method performs a right shift.
(Computes floor(this * 2^{n}).)
n
 shift distance, in bits.
this << n
shiftRight(int)
public BigInteger shiftRight(int n)
(this >> n)
. Sign
extension is performed. The shift distance, n
, may be
negative, in which case this method performs a left shift.
(Computes floor(this / 2^{n}).)
n
 shift distance, in bits.
this >> n
shiftLeft(int)
public BigInteger and(BigInteger val)
(this & val)
. (This
method returns a negative BigInteger if and only if this and val are
both negative.)
val
 value to be AND'ed with this BigInteger.
this & val
public BigInteger or(BigInteger val)
(this  val)
. (This method
returns a negative BigInteger if and only if either this or val is
negative.)
val
 value to be OR'ed with this BigInteger.
this  val
public BigInteger xor(BigInteger val)
(this ^ val)
. (This method
returns a negative BigInteger if and only if exactly one of this and
val are negative.)
val
 value to be XOR'ed with this BigInteger.
this ^ val
public BigInteger not()
(~this)
. (This method
returns a negative value if and only if this BigInteger is
nonnegative.)
~this
public BigInteger andNot(BigInteger val)
(this & ~val)
. This
method, which is equivalent to and(val.not())
, is provided as
a convenience for masking operations. (This method returns a negative
BigInteger if and only if this
is negative and val
is
positive.)
val
 value to be complemented and AND'ed with this BigInteger.
this & ~val
public boolean testBit(int n)
true
if and only if the designated bit is set.
(Computes ((this & (1<<n)) != 0)
.)
n
 index of bit to test.
true
if and only if the designated bit is set.
ArithmeticException
 n
is negative.public BigInteger setBit(int n)
(this  (1<<n))
.)
n
 index of bit to set.
this  (1<<n)
ArithmeticException
 n
is negative.public BigInteger clearBit(int n)
(this & ~(1<<n))
.)
n
 index of bit to clear.
this & ~(1<<n)
ArithmeticException
 n
is negative.public BigInteger flipBit(int n)
(this ^ (1<<n))
.)
n
 index of bit to flip.
this ^ (1<<n)
ArithmeticException
 n
is negative.public int getLowestSetBit()
(this==0? 1 : log2(this & this))
.)
public int bitLength()
(ceil(log2(this < 0 ? this : this+1)))
.)
public int bitCount()
public boolean isProbablePrime(int certainty)
true
if this BigInteger is probably prime,
false
if it's definitely composite. If
certainty
is <= 0
, true
is
returned.
certainty
 a measure of the uncertainty that the caller is
willing to tolerate: if the call returns true
the probability that this BigInteger is prime exceeds
(1  1/2^{certainty}). The execution time of
this method is proportional to the value of this parameter.
true
if this BigInteger is probably prime,
false
if it's definitely composite.public int compareTo(BigInteger val)
(x.compareTo(y)
<op> 0)
, where
<op> is one of the six comparison operators.
compareTo
in interface Comparable<BigInteger>
val
 BigInteger to which this BigInteger is to be compared.
val
.public boolean equals(Object x)
equals
in class Object
x
 Object to which this BigInteger is to be compared.
true
if and only if the specified Object is a
BigInteger whose value is numerically equal to this BigInteger.Object.hashCode()
,
Hashtable
public BigInteger min(BigInteger val)
val
.
val
 value with which the minimum is to be computed.
val
. If they are equal, either may be returned.public BigInteger max(BigInteger val)
val
.
val
 value with which the maximum is to be computed.
val
. If they are equal, either may be returned.public int hashCode()
hashCode
in class Object
Object.equals(java.lang.Object)
,
Hashtable
public String toString(int radix)
Character.MIN_RADIX
to Character.MAX_RADIX
inclusive,
it will default to 10 (as is the case for
Integer.toString
). The digittocharacter mapping
provided by Character.forDigit
is used, and a minus
sign is prepended if appropriate. (This representation is
compatible with the (String,
int)
constructor.)
radix
 radix of the String representation.
Integer.toString(int, int)
,
Character.forDigit(int, int)
,
BigInteger(java.lang.String, int)
public String toString()
Character.forDigit
is used, and a minus sign is
prepended if appropriate. (This representation is compatible
with the (String)
constructor, and
allows for String concatenation with Java's + operator.)
toString
in class Object
Character.forDigit(int, int)
,
BigInteger(java.lang.String)
public byte[] toByteArray()
(ceil((this.bitLength() +
1)/8))
. (This representation is compatible with the
(byte[])
constructor.)
BigInteger(byte[])
public int intValue()
int
. This
conversion is analogous to a narrowing
primitive conversion from long
to
int
as defined in the Java Language
Specification: if this BigInteger is too big to fit in an
int
, only the loworder 32 bits are returned.
Note that this conversion can lose information about the
overall magnitude of the BigInteger value as well as return a
result with the opposite sign.
intValue
in class Number
int
.public long longValue()
long
. This
conversion is analogous to a narrowing
primitive conversion from long
to
int
as defined in the Java Language
Specification: if this BigInteger is too big to fit in a
long
, only the loworder 64 bits are returned.
Note that this conversion can lose information about the
overall magnitude of the BigInteger value as well as return a
result with the opposite sign.
longValue
in class Number
long
.public float floatValue()
float
. This
conversion is similar to the narrowing
primitive conversion from double
to
float
defined in the Java Language
Specification: if this BigInteger has too great a magnitude
to represent as a float
, it will be converted to
Float.NEGATIVE_INFINITY
or Float.POSITIVE_INFINITY
as appropriate. Note that even when
the return value is finite, this conversion can lose
information about the precision of the BigInteger value.
floatValue
in class Number
float
.public double doubleValue()
double
. This
conversion is similar to the narrowing
primitive conversion from double
to
float
defined in the Java Language
Specification: if this BigInteger has too great a magnitude
to represent as a double
, it will be converted to
Double.NEGATIVE_INFINITY
or Double.POSITIVE_INFINITY
as appropriate. Note that even when
the return value is finite, this conversion can lose
information about the precision of the BigInteger value.
doubleValue
in class Number
double
.

Java™ Platform Standard Ed. 6 

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