Class Qp

Represents p-adic numbers that are rational, denoted ℚ⊂ℚₚ.

Inheritance
object
Q
Qp
Implements
Inherited Members
Namespace: MathLib
Assembly: MathLib.dll
Syntax
public class Qp : Q, IEquatable<Q>, IComparable<Q>
Remarks

In mathematics, the set of p-adic numbers ℚₚ is the completion of the rational numbers ℚ with respect to the p-adic norm. This completion includes both rational numbers and p-adic irrationals (limits of infinite p-adic expansions).

However, in this library, the Qp class is restricted to representing only the rational numbers within ℚₚ (which can be finitely represented in a computer). Consequently, it handles p-adic numbers whose expansions are finite or ultimately periodic, corresponding exactly to elements of ℚ

This means that while ℚₚ is uncountable and includes numbers that cannot be expressed as ratios of integers (p-adic irrationals), our Qp class cannot represent these elements. It focuses on rational numbers and their representations in p-adic form.

Examples

A 5-adic expansion for the rational number -4/3 is:

-4/3 = …2222222212₅

Constructors

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Qp(BaseInt, BaseInt, int)

Constructor for a p-adic number for the specified preperiodic and periodic parts, and an optional first exponent.

Declaration
public Qp(BaseInt pAdicPreperiodic, BaseInt pAdicPeriodic, int firstExponent = 0)
Parameters
Type Name Description
BaseInt pAdicPreperiodic

The p-adic preperiodic part of the number.
BaseInt pAdicPeriodic

The p-adic periodic part of the number.
int firstExponent

The first exponent of the number (Default value = 0).
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Qp(Q, int)

Constructor for a p-adic number.

Declaration
public Qp(Q q, int base_)
Parameters
Type Name Description
Q q

The rational number to represent as a p-adic number.
int base_

The prime base of the p-adic number.
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Qp(BigInteger, BigInteger, int)

Declaration
public Qp(BigInteger Numerator, BigInteger Denominator, int base_)
Parameters
Type Name Description
BigInteger Numerator
BigInteger Denominator
int base_

Properties

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Base

Returns the prime base of this p-adic number.

Declaration
public int Base { get; }
Property Value
Type Description
int
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FirstExponent

Declaration
public int FirstExponent { get; }
Property Value
Type Description
int
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Generator

A reverse generator that can be used to generate the coefficients of the p-adic number, in reverse order (least significant digit first).

Declaration
public Qb Generator { get; }
Property Value
Type Description
Qb
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NaN

Returns a NaN Qp instance.

Declaration
public static Qp NaN { get; }
Property Value
Type Description
Qp

Methods

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Coefficients()

Declaration
public IEnumerable<int> Coefficients()
Returns
Type Description
IEnumerable<int>
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FindKQ(Q, int)

Decomposes a rational number q into an integer part k and a fractional remainder based on the specified modulus.

Declaration
public static (int k, Q q) FindKQ(Q q, int modulus)
Parameters
Type Name Description
Q q

The rational number to decompose, represented as an instance of Q.
int modulus

The modulus (or base) for the decomposition, which must be a positive integer greater than 1.
Returns
Type Description
(int k, Q q)

A tuple containing:

  • k: An integer representing the integer component of the decomposition, in the range [0, modulus).
  • q: The remaining fractional part of the decomposition, represented as a rational number Q.
Remarks

This method computes the first coefficient k in the p-adic expansion of the rational number q with respect to the specified modulus modulus (which serves as the base of the p-adic expansion). The remainder is the fractional part of the number after subtracting the integer part k and dividing by modulus.

Mathematically, this method solves the equation:

q = k + modulus × remainder

where k is an integer in the range [0, modulus) and remainder is the fractional part of q after division by modulus.

The method uses modular arithmetic to compute k efficiently by solving the congruence:

k ≡ q.Numerator × (q.Denominator)⁻¹ (mod modulus)

where q.Denominator⁻¹ is the modular inverse of q.Denominator with respect to modulus.

Examples

This example demonstrates decomposing a rational number with respect to a modulus:

Q q = new Q(1, 17);
(int k, Q remainder) = FindKQ(q, 5);
Console.WriteLine($"k = {k}, remainder = {remainder}");
// Output: k = 3, remainder = -10 / 17
Exceptions
Type Condition
ArgumentException

Thrown if q's denominator and modulus are not coprime (i.e., their greatest common divisor is not 1).
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PadicCoeffs(Q, int, bool)

Declaration
public static IEnumerable<int> PadicCoeffs(Q q, int base_, bool yieldDelimiters = false)
Parameters
Type Name Description
Q q
int base_
bool yieldDelimiters
Returns
Type Description
IEnumerable<int>
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ReciprocalCoefficients(BigInteger, int)

Generates the p-adic coefficients of the reciprocal of n in the base base_. Also returns a remainder with each coefficient.

Declaration
public static IEnumerable<(int coefficient, BigInteger remainder)> ReciprocalCoefficients(BigInteger n, int base_)
Parameters
Type Name Description
BigInteger n

The number whose reciprocal's p-adic expansion is computed.
int base_

The prime base of the p-adic expansion.
Returns
Type Description
IEnumerable<(int coefficient, BigInteger remainder)>

An enumerable tuple sequence (coefficient, remainder).
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ToString()

Default string representation of the p-adic number.

Declaration
public override string ToString()
Returns
Type Description
string
Overrides
Q.ToString()
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ToStringExpanded(int)

Returns a string for the p-adic number in expanded form.

Declaration
public string ToStringExpanded(int coefficientCount = 16)
Parameters
Type Name Description
int coefficientCount

The number of coefficients to include
Returns
Type Description
string
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ToStringPeriodic()

Declaration
public string ToStringPeriodic()
Returns
Type Description
string
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Valuation()

p-adic valuation of the current p-adic number.

Declaration
public int Valuation()
Returns
Type Description
int

Valuation of the number relative to Base.

Implements

IEquatable<T>
IComparable<T>