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Class Description
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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.
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 right-shift with a negative
shift distance results in a left shift, and vice-versa. 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 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
"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, 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 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.
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Class Variables
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ZERO
public static final BigInteger
The BigInteger constant zero.
ONE
public static final BigInteger
The BigInteger constant one.
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Instance Variables
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None declared in this class.
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Constructors
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BigInteger
public BigInteger(
byte[] val
)
Translates a byte array containing the two's-complement binary
representation of a BigInteger into a BigInteger. The input array is
assumed to be in big-endian byte-order: the most significant
byte is in the zeroth element.
BigInteger
public BigInteger(
int signum,
byte[] magnitude
)
Translates the sign-magnitude representation of a BigInteger into a
BigInteger. The sign is represented as an integer signum value: -1 for
negative, 0 for zero, or 1 for positive. The magnitude is a byte array
in big-endian byte-order: the most significant byte is in the
zeroth element. A zero-length magnitude array is permissible, and will
result in in a BigInteger value of 0, whether signum is -1, 0 or 1.
BigInteger
public BigInteger(
int bitLength,
int certainty,
Random rnd
)
Constructs a randomly generated positive BigInteger that is probably
prime, with the specified bitLength.
It is recommended that the probablePrime
method be used in preference to this constructor unless there
is a compelling need to specify a certainty.
BigInteger
public BigInteger(
int numBits,
Random rnd
)
Constructs a randomly generated BigInteger, uniformly distributed over
the range 0 to (2numBits - 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 non-negative BigInteger.
BigInteger
public BigInteger(
String val
)
Translates the decimal String representation of a BigInteger into a
BigInteger. The String representation consists of an optional minus
sign followed by a sequence of one or more decimal digits. The
character-to-digit mapping is provided by Character.digit.
The String may not contain any extraneous characters (whitespace, for
example).
BigInteger
public BigInteger(
String val,
int radix
)
Translates the String representation of a BigInteger in the specified
radix into a BigInteger. The String representation consists of an
optional minus sign followed by a sequence of one or more digits in the
specified radix. The character-to-digit mapping is provided by
Character.digit. The String may not contain any extraneous
characters (whitespace, for example).
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Class Methods
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probablePrime
public static BigInteger probablePrime(
int bitLength,
Random rnd
)
Returns a positive BigInteger that is probably prime, with the
specified bitLength. The probability that a BigInteger returned
by this method is composite does not exceed 2-100.
valueOf
public static BigInteger valueOf(
long val
)
Returns a BigInteger whose value is equal to that of the
specified long. This "static factory method" is
provided in preference to a (long) constructor
because it allows for reuse of frequently used BigIntegers.
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Instance Methods
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abs
public BigInteger abs(
)
Returns a BigInteger whose value is the absolute value of this
BigInteger.
add
public BigInteger add(
BigInteger val
)
Returns a BigInteger whose value is (this + val).
and
public BigInteger and(
BigInteger val
)
Returns a BigInteger whose value is (this & val). (This
method returns a negative BigInteger if and only if this and val are
both negative.)
andNot
public BigInteger andNot(
BigInteger val
)
Returns a BigInteger whose value is (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.)
bitCount
public int bitCount(
)
Returns the number of bits in the two's complement representation
of this BigInteger that differ from its sign bit. This method is
useful when implementing bit-vector style sets atop BigIntegers.
bitLength
public int bitLength(
)
Returns the number of bits in the minimal two's-complement
representation of this BigInteger, excluding a sign bit.
For positive BigIntegers, this is equivalent to the number of bits in
the ordinary binary representation. (Computes
(ceil(log2(this < 0 ? -this : this+1))).)
clearBit
public BigInteger clearBit(
int n
)
Returns a BigInteger whose value is equivalent to this BigInteger
with the designated bit cleared.
(Computes (this & ~(1<<n)).)
compareTo
public int compareTo(
BigInteger val
)
Compares this BigInteger with the specified BigInteger. This method is
provided in preference to individual methods for each of the six
boolean comparison operators (<, ==, >, >=, !=, <=). The
suggested idiom for performing these comparisons is:
(x.compareTo(y) <op> 0),
where <op> is one of the six comparison operators.
compareTo
public int compareTo(
Object o
)
Compares this BigInteger with the specified Object. If the Object is a
BigInteger, this method behaves like compareTo(BigInteger).
Otherwise, it throws a ClassCastException (as BigIntegers are
comparable only to other BigIntegers).
divide
public BigInteger divide(
BigInteger val
)
Returns a BigInteger whose value is (this / val).
divideAndRemainder
public BigInteger[] divideAndRemainder(
BigInteger val
)
Returns an array of two BigIntegers containing (this / val)
followed by (this % val).
doubleValue
public double doubleValue(
)
Converts this BigInteger to a 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
NEGATIVE_INFINITY or 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.
equals
public boolean equals(
Object x
)
Compares this BigInteger with the specified Object for equality.
flipBit
public BigInteger flipBit(
int n
)
Returns a BigInteger whose value is equivalent to this BigInteger
with the designated bit flipped.
(Computes (this ^ (1<<n)).)
floatValue
public float floatValue(
)
Converts this BigInteger to a 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
NEGATIVE_INFINITY or 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.
gcd
public BigInteger gcd(
BigInteger val
)
Returns a BigInteger whose value is the greatest common divisor of
abs(this) and abs(val). Returns 0 if
this==0 && val==0.
getLowestSetBit
public int getLowestSetBit(
)
Returns the index of the rightmost (lowest-order) one bit in this
BigInteger (the number of zero bits to the right of the rightmost
one bit). Returns -1 if this BigInteger contains no one bits.
(Computes (this==0? -1 : log2(this & -this)).)
hashCode
public int hashCode(
)
Returns the hash code for this BigInteger.
intValue
public int intValue(
)
Converts this BigInteger to an 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 low-order 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.
isProbablePrime
public boolean isProbablePrime(
int certainty
)
Returns true if this BigInteger is probably prime,
false if it's definitely composite. If
certainty is <= 0, true is
returned.
longValue
public long longValue(
)
Converts this BigInteger to a 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 low-order 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.
max
public BigInteger max(
BigInteger val
)
Returns the maximum of this BigInteger and val.
min
public BigInteger min(
BigInteger val
)
Returns the minimum of this BigInteger and val.
mod
public BigInteger mod(
BigInteger m
)
Returns a BigInteger whose value is (this mod m). This method
differs from remainder in that it always returns a
non-negative BigInteger.
modInverse
public BigInteger modInverse(
BigInteger m
)
Returns a BigInteger whose value is (this-1 mod m).
modPow
public BigInteger modPow(
BigInteger exponent,
BigInteger m
)
Returns a BigInteger whose value is
(thisexponent mod m). (Unlike pow, this
method permits negative exponents.)
multiply
public BigInteger multiply(
BigInteger val
)
Returns a BigInteger whose value is (this * val).
negate
public BigInteger negate(
)
Returns a BigInteger whose value is (-this).
not
public BigInteger not(
)
Returns a BigInteger whose value is (~this). (This method
returns a negative value if and only if this BigInteger is
non-negative.)
or
public BigInteger or(
BigInteger val
)
Returns a BigInteger whose value is (this | val). (This method
returns a negative BigInteger if and only if either this or val is
negative.)
pow
public BigInteger pow(
int exponent
)
Returns a BigInteger whose value is (thisexponent).
Note that exponent is an integer rather than a BigInteger.
remainder
public BigInteger remainder(
BigInteger val
)
Returns a BigInteger whose value is (this % val).
setBit
public BigInteger setBit(
int n
)
Returns a BigInteger whose value is equivalent to this BigInteger
with the designated bit set. (Computes (this | (1<<n)).)
shiftLeft
public BigInteger shiftLeft(
int n
)
Returns a BigInteger whose value is (this << n).
The shift distance, n, may be negative, in which case
this method performs a right shift.
(Computes floor(this * 2n).)
shiftRight
public BigInteger shiftRight(
int n
)
Returns a BigInteger whose value is (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 / 2n).)
signum
public int signum(
)
Returns the signum function of this BigInteger.
subtract
public BigInteger subtract(
BigInteger val
)
Returns a BigInteger whose value is (this - val).
testBit
public boolean testBit(
int n
)
Returns true if and only if the designated bit is set.
(Computes ((this & (1<<n)) != 0).)
toByteArray
public byte[] toByteArray(
)
Returns a byte array containing the two's-complement
representation of this BigInteger. The byte array will be in
big-endian byte-order: the most significant byte is in
the zeroth element. The array will contain the minimum number
of bytes required to represent this BigInteger, including at
least one sign bit, which is (ceil((this.bitLength() +
1)/8)). (This representation is compatible with the
(byte[]) constructor.)
toString
public String toString(
)
Returns the decimal String representation of this BigInteger.
The digit-to-character mapping provided by
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
public String toString(
int radix
)
Returns the String representation of this BigInteger in the
given radix. If the radix is outside the range from MIN_RADIX to MAX_RADIX inclusive,
it will default to 10 (as is the case for
Integer.toString). The digit-to-character mapping
provided by Character.forDigit is used, and a minus
sign is prepended if appropriate. (This representation is
compatible with the (String,
int) constructor.)
xor
public BigInteger xor(
BigInteger val
)
Returns a BigInteger whose value is (this ^ val). (This method
returns a negative BigInteger if and only if exactly one of this and
val are negative.)
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