Class BigIntegerExtensions

Provides a set of extension methods for the BigInteger struct, enabling additional functionality such as determining whether a number is odd, computing its absolute value, and calculating its bit length. It also provides the Extended Euclidean algorithm for computing GCD and Bézout coefficients.

Inheritance
object
BigIntegerExtensions
Inherited Members
Namespace: MathLib
Assembly: MathLib.dll
Syntax
public static class BigIntegerExtensions

Methods

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Abs(BigInteger)

Absolute value of the specified integer value.

Declaration
public static BigInteger Abs(this BigInteger integer)
Parameters
Type Name Description
BigInteger integer

The BigInteger value whose absolute value is to be computed.
Returns
Type Description
BigInteger

A BigInteger representing the absolute value of the input.
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Coprime(BigInteger, BigInteger)

Indicates whether two specified BigInteger values are coprime.

Declaration
public static bool Coprime(this BigInteger a, BigInteger b)
Parameters
Type Name Description
BigInteger a

The first BigInteger value.
BigInteger b

The second BigInteger value.
Returns
Type Description
bool

true iff a and b are coprime.
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ExtendedEuclideanAlgorithm(BigInteger, BigInteger)

Implements the extended Euclidean algorithm to compute the greatest common divisor (GCD) of two BigInteger values and the corresponding Bézout coefficients.

Declaration
public static (BigInteger gcd, BigInteger x, BigInteger y) ExtendedEuclideanAlgorithm(BigInteger a, BigInteger b)
Parameters
Type Name Description
BigInteger a

The first BigInteger.
BigInteger b

The second BigInteger.
Returns
Type Description
(BigInteger gcd, BigInteger x, BigInteger y)

A tuple containing three BigInteger values:

  • The greatest common divisor (gcd) of a and b.
  • The Bézout coefficient 'x' (corresponding to a in Bézout's identity).
  • The Bézout coefficient 'y' (corresponding to b in Bézout's identity).
Remarks

For two integers a and b, this method computes the GCD, denoted as gcd(a, b), and finds integers 'x' and 'y' such that Bézout's identity is satisfied:

a·x + b· y = gcd(a, b)
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IsDivisibleBy(BigInteger, BigInteger)

Indicates whether the specified BigInteger value is divisible by the specified divisor.

Declaration
public static bool IsDivisibleBy(this BigInteger integer, BigInteger divisor)
Parameters
Type Name Description
BigInteger integer

The BigInteger value to check.
BigInteger divisor

The divisor to check against.
Returns
Type Description
bool

true iff the value is divisible by the divisor.
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IsOdd(BigInteger)

Indicates whether the specified BigInteger value is odd.

Declaration
public static bool IsOdd(this BigInteger integer)
Parameters
Type Name Description
BigInteger integer

The BigInteger value to check.
Returns
Type Description
bool

true iff integer is odd.
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IsPowerOf(BigInteger, int)

Indicates whether the specified BigInteger value is a power of the specified exponent.

Declaration
public static bool IsPowerOf(this BigInteger integer, int exponent)
Parameters
Type Name Description
BigInteger integer

The BigInteger value to check.
int exponent

The exponent to check against.
Returns
Type Description
bool

true iff the value is a power of the exponent.
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LCM(BigInteger, BigInteger)

Returns the Least Common Multiple of two integers

Declaration
public static BigInteger LCM(this BigInteger first, BigInteger second)
Parameters
Type Name Description
BigInteger first

The first integer
BigInteger second

The second integer
Returns
Type Description
BigInteger

LCM of first and second
Examples

LCM(4, 6) = 12, since 12 is both a multiple of 4 (43) and a multiple of 6 (62)

Exceptions
Type Condition
ArgumentOutOfRangeException

Thrown if first or second is zero.
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Length(BigInteger, int)

Returns the number of coefficients (or digits) required to represent this
BigInteger in a specified base.

Declaration
public static int Length(this BigInteger integer, int base_)
Parameters
Type Name Description
BigInteger integer

The BigInteger to calculate the coefficient count for.
int base_

The base in which this BigInteger will be represented. Must be greater than 1 for meaningful results.
Returns
Type Description
int

The number of coefficients required to represent this BigInteger in base base_.

0 if base_ is less than 2.
Remarks

If this BigInteger is 0, the result is always 0, regardless of the base.

If this BigInteger is negative, the result will be the same as for its absolute value.

If the base base_ is less than 2, this method returns 0, following the behavior of Log(BigInteger, double).

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Mod(BigInteger, BigInteger)

Computes the modulus of a BigInteger in the same way as in languages like Python, Haskell, Julia and Matlab. The result follows the mathematical definition of modulus where the remainder always has the same sign as the divisor, ensuring predictable results for both positive and negative values.

Declaration
public static BigInteger Mod(this BigInteger integer, BigInteger modulus)
Parameters
Type Name Description
BigInteger integer

The BigInteger value to compute the modulus for.
BigInteger modulus

The modulus value. Must be non-zero.
Returns
Type Description
BigInteger

The remainder when integer is divided by modulus, adjusted to follow the same sign as modulus.
Remarks

The result of the modulus operation is adjusted based on the sign of modulus:

When modulus is positive, the result is in the range [0, modulus)

When modulus is negative, the result is in the range (modulus, 0].

This method provides consistent behavior for modulus operations, similar to how it is implemented in Python, Haskell, Julia, and Matlab.

Examples

Examples:

BigInteger result1 = new BigInteger(10).Mod(3);  // 1
BigInteger result2 = new BigInteger(-10).Mod(3); // 2
BigInteger result3 = new BigInteger(10).Mod(-3); // -2
BigInteger result4 = new BigInteger(-10).Mod(-3); // -1
Exceptions
Type Condition
DivideByZeroException

Thrown when modulus is zero.
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ModMinAbs(BigInteger, BigInteger)

Modulo operation yielding the remainder (positive or negative) with the smallest absolute value, preserving congruence under the specified modulus.

Declaration
public static BigInteger ModMinAbs(this BigInteger integer, BigInteger modulus)
Parameters
Type Name Description
BigInteger integer

Dividend, an integer.
BigInteger modulus

Modulus, a positive integer.
Returns
Type Description
BigInteger

The integer r closest to zero for which there exists an integer k such that n = k × mod + r, and -mod/2 ≤ r ≤ mod/2.
Remarks

For both positive and negative integer, the result minimizes |r|, providing a "balanced" or symmetric result. This is useful in contexts where closeness to zero is desired, such as balanced systems or modular arithmetic with minimal magnitude deviation.

For example:

  • 4 Mod 7 yields -3 instead of 4, as -3 is closer to zero.
  • 9 Mod 7 yields 2, as 2 is already closest to zero.
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ModularInverse(BigInteger, BigInteger)

Computes the modular inverse of a number integer under a given modulus modulus.

Declaration
public static BigInteger ModularInverse(this BigInteger integer, BigInteger modulus)
Parameters
Type Name Description
BigInteger integer

The number for which the modular inverse is to be computed. Must be greater than or equal to 1.
BigInteger modulus

The modulus under which the inverse is computed. Must be greater than or equal to 2.
Returns
Type Description
BigInteger

The modular inverse of integer modulo modulus. The result is normalized to be within the range [0, modulus).
Remarks

The modular inverse of integer modulo modulus is a number x such that:

integer ⋅ x ≡ 1 (mod modulus)

This method uses the Extended Euclidean Algorithm to compute the inverse.

The result of this method is always in the range [0, modulus), as it is normalized by applying the modulus operation.

Examples
BigInteger result1 = new BigInteger(3).ModularInverse(7);  // 5
BigInteger result2 = new BigInteger(2).ModularInverse(11); // 6
Exceptions
Type Condition
ArgumentOutOfRangeException

Thrown if integer is less than 1 or if modulus is less than 2.
ArgumentException

Thrown if integer and modulus are not coprime, i.e., if the greatest common divisor of the two is not 1.
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Parse(string, int)

Parses the string representation of a number in the specified base to its BigInteger equivalent.

Declaration
public static BigInteger Parse(string input, int fromBase)
Parameters
Type Name Description
string input

The string representation of the number to parse. It may start with a '-' to indicate a negative number.
int fromBase

The base of the input number, ranging from 2 to 36.
Returns
Type Description
BigInteger

A BigInteger representation of the parsed value.
Examples
Console.WriteLine(Parse("-10.011", 2));  //outputs "-19/8"
Exceptions
Type Condition
ArgumentException

Thrown if fromBase is not within 2 to 36.