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SymPy 1.11.dev documentation
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SymPy 1.11.dev documentation
  • SymPy Tutorial
    • Preliminaries
    • Introduction
    • Gotchas
    • SymPy Features
      • Basic Operations
      • Printing
      • Simplification
      • Calculus
      • Solvers
      • Matrices
      • Advanced Expression Manipulation
    • What’s Next
  • SymPy Guides
    • Getting Started
      • Installation
    • Contributing to SymPy
      • Development Environment Setup
      • Dependencies
      • Build the Documentation
      • Debugging
      • SymPy Docstrings Style Guide
      • General Guidelines
      • Types of Documentation
      • Narrative Documentation Guidelines
      • Documentation using Markdown
      • Best Practices for Writing Documentation
      • Making a Contribution
      • Deprecation Policy
    • Assumptions
    • Symbolic and fuzzy booleans
  • Explanation
    • Gotchas and Pitfalls
    • SymPy Special Topics
      • Finite Difference Approximations to Derivatives
      • Classification of SymPy objects
    • List of active deprecations
  • Reference Documentation
    • Basics
      • Assumptions
        • Ask
        • Assume
        • Refine
        • Predicates
      • Calculus
      • Combinatorics
        • Partitions
        • Permutations
        • Permutation Groups
        • Polyhedron
        • Prufer Sequences
        • Subsets
        • Gray Code
        • Named Groups
        • Utilities
        • Group constructors
        • Test Utilities
        • Tensor Canonicalization
        • Finitely Presented Groups
        • Polycyclic Groups
      • Functions
        • Elementary
        • sympy.functions.elementary.complexes
        • sympy.functions.elementary.trigonometric
        • Trigonometric Functions
        • Trigonometric Inverses
        • sympy.functions.elementary.hyperbolic
        • Hyperbolic Functions
        • Hyperbolic Inverses
        • sympy.functions.elementary.integers
        • sympy.functions.elementary.exponential
        • sympy.functions.elementary.piecewise
        • sympy.functions.elementary.miscellaneous
        • Combinatorial
        • Enumeration
        • Special
      • Integrals
        • Computing Integrals using Meijer G-Functions
        • The G-Function Integration Theorems
        • The Inverse Laplace Transform of a G-function
        • Implemented G-Function Formulae
        • Internal API Reference
        • Integrals
      • Series
        • Series Expansions
        • Sequences
        • Fourier Series
        • Formal Power Series
        • Limits of Sequences
      • Simplify
        • Simplify
        • Hypergeometric Expansion
        • Hongguang Fu’s Trigonometric Simplification
      • Solvers
        • Diophantine
        • Inequality Solvers
        • ODE
        • PDE
        • Solvers
        • Solveset
      • abc
      • Algebras
      • Concrete
      • Core
      • Discrete
      • Numerical Evaluation
      • Numeric Computation
      • Term Rewriting
    • Code Generation
      • Introduction
      • Code printers (sympy.printing)
      • Codegen (sympy.utilities.codegen)
      • Autowrap
      • Classes and functions for rewriting expressions (sympy.codegen.rewriting)
      • Tools for simplifying expressions using approximations (sympy.codegen.approximations)
      • Classes for abstract syntax trees (sympy.codegen.ast)
      • Special C math functions (sympy.codegen.cfunctions)
      • C specific AST nodes (sympy.codegen.cnodes)
      • C++ specific AST nodes (sympy.codegen.cxxnodes)
      • Fortran specific AST nodes (sympy.codegen.fnodes)
      • Algorithms (sympy.codegen.algorithms)
      • Python utilities (sympy.codegen.pyutils)
      • C utilities (sympy.codegen.cutils)
      • Fortran utilities (sympy.codegen.futils)
    • Logic
      • Logic
      • Sets
    • Matrices
      • Matrices
        • Matrices (linear algebra)
        • Common Matrices
        • Dense Matrices
        • Sparse Matrices
        • Sparse Tools
        • Immutable Matrices
        • Matrix Expressions
        • Matrix Normal Forms
      • Tensor
        • N-dim array
        • N-dim array expressions
        • Indexed Objects
        • Methods
        • Tensor
        • Tensor Operators
      • Vector
        • Introduction
        • Basic Implementation details
        • More about Coordinate Systems
        • Scalar and Vector Field Functionality
        • General examples of usage
        • Applications of Vector Integrals
        • Essential Classes in sympy.vector (docstrings)
        • Orienter classes (docstrings)
        • Essential Functions in sympy.vector (docstrings)
    • Number Theory
      • Ntheory Class Reference
      • Ntheory Functions Reference
    • Physics
      • Hydrogen Wavefunctions
      • Matrices
      • Pauli Algebra
      • Quantum Harmonic Oscillator in 1-D
      • Quantum Harmonic Oscillator in 3-D
      • Second Quantization
      • Wigner Symbols
      • Unit systems
        • Philosophy behind unit systems
        • Examples
        • Dimensions and dimension systems
        • Unit prefixes
        • Units and unit systems
        • Physical quantities
      • High energy physics
      • The Physics Vector Module
        • Vector & ReferenceFrame
        • Vector: Kinematics
        • Potential Issues/Advanced Topics/Future Features in Physics/Vector Module
        • Scalar and Vector Field Functionality
        • Essential Classes
        • Kinematics (Docstrings)
        • Printing (Docstrings)
        • Essential Functions (Docstrings)
        • Docstrings for basic field functions
      • Classical Mechanics
        • Masses, Inertias, Particles and Rigid Bodies in Physics/Mechanics
        • Kane’s Method in Physics/Mechanics
        • Lagrange’s Method in Physics/Mechanics
        • Symbolic Systems in Physics/Mechanics
        • Linearization in Physics/Mechanics
          • Nonminimal Coordinates Pendulum
        • Examples for Physics/Mechanics
          • A rolling disc
            • A rolling disc, with Kane’s method
            • A rolling disc, with Kane’s method and constraint forces
            • A rolling disc using Lagrange’s Method
          • A bicycle
          • Nonminimal Coordinates Pendulum
          • Multi Degree of Freedom Holonomic System
        • Potential Issues/Advanced Topics/Future Features in Physics/Mechanics
        • References for Physics/Mechanics
        • Autolev Parser
        • SymPy Mechanics for Autolev Users
        • Masses, Inertias & Particles, RigidBodys (Docstrings)
        • Kane’s Method & Lagrange’s Method (Docstrings)
        • SymbolicSystem (Docstrings)
        • Linearization (Docstrings)
        • Expression Manipulation (Docstrings)
        • Printing (Docstrings)
        • Body (Docstrings)
        • Joint (Docstrings)
      • Quantum Mechanics
        • Anticommutator
        • Clebsch-Gordan Coefficients
        • Commutator
        • Constants
        • Dagger
        • Inner Product
        • Tensor Product
        • Cartesian Operators and States
        • Hilbert Space
        • Operator
        • Operator/State Helper Functions
        • Qapply
        • Represent
        • Spin
        • State
        • Circuit Plot
        • Gates
        • Grover’s Algorithm
        • QFT
        • Qubit
        • Shor’s Algorithm
        • Particle in a Box
      • Optics Module
        • Gaussian Optics
        • Medium
        • Polarization
        • Utilities
        • Waves
      • Control Module
        • Control
        • Control API
        • Control System Plots
      • Continuum Mechanics
        • Beam (Docstrings)
        • Solving Beam Bending Problems using Singularity Functions
    • Utilities
      • Testing
        • pytest
        • Randomised Testing
        • Run Tests
      • Utilities
        • Autowrap Module
        • Codegen
        • Decorator
        • Enumerative
        • Exceptions and Warnings
        • Iterables
        • Lambdify
        • Memoization
        • Miscellaneous
        • PKGDATA
        • Source Code Inspection
        • Timing Utilities
      • Interactive
      • Parsing
      • Printing
    • Topics
      • Geometry
        • Entities
        • Utils
        • Points
        • Lines
        • Curves
        • Ellipses
        • Polygons
        • Plane
      • Holonomic
        • About Holonomic Functions
        • Representation of holonomic functions in SymPy
        • Operations on holonomic functions
        • Converting other representations to holonomic
        • Uses and Current limitations
        • Internal API
      • Lie Algebra
      • Polynomial Manipulation
        • Basic functionality of the module
        • Examples from Wester’s Article
        • Polynomials Manipulation Module Reference
        • AGCA - Algebraic Geometry and Commutative Algebra Module
        • Introducing the Domains of the poly module
        • Reference docs for the Poly Domains
        • Internals of the Polynomial Manipulation Module
        • Series Manipulation using Polynomials
        • Literature
        • Poly solvers
        • Introducing the domainmatrix of the poly module
        • Number Fields
      • Category Theory
      • Cryptography
      • Differential Geometry
      • Plotting
      • Stats
  • Miscellaneous
    • About
    • Citing SymPy
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    • SymPy Papers
    • Planet SymPy
    • SymPy logos
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Number Theory#

Contents

  • Ntheory Class Reference
  • Ntheory Functions Reference
    • ECM function
    • Examples
    • QS function
    • Examples
Next
Ntheory Class Reference
Previous
Essential Functions in sympy.vector (docstrings)
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Last updated on Feb 17, 2022