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PARI/GP is a widely used computer algebra system designed for fast computations in number theory (factorizations, algebraic number theory, elliptic curves, modular forms, L functions...), but also contains a large number of other useful functions to compute with mathematical entities such as matrices, polynomials, power series, algebraic numbers etc., and a lot of transcendental functions. PARI is also available as a C library to allow for faster computations.
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User’s Guide
to
PARI / GP
(version 2.9.2)
The PARI Group
Institut de Math´ematiques de Bordeaux, UMR 5251 du CNRS.
Universit´e de Bordeaux, 351 Cours de la Lib´eration
F-33405 TALENCE Cedex, FRANCE
e-mail: pari@math.u-bordeaux.fr
Home Page:
http://pari.math.u-bordeaux.fr/

Copyright
c
2000–2016 The PARI Group
Permission is granted to make and distribute verbatim copies of this manual provided the copyright
notice and this permission notice are preserved on all copies.
Permission is granted to copy and distribute modified versions, or translations, of this manual
under the conditions for verbatim copying, provided also that the entire resulting derived work is
distributed under the terms of a permission notice identical to this one.
PARI/GP is Copyright
c
2000–2016 The PARI Group
PARI/GP is free software; you can redistribute it and/or modify it under the terms of the GNU
General Public License as published by the Free Software Foundation. It is distributed in the hope
that it will be useful, but WITHOUT ANY WARRANTY WHATSOEVER.

Table of Contents
Chapter 1: Overview of the PARI system . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2 Multiprecision kernels / Portability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 The PARI types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.4 The PARI philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Operations and functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
Chapter 2: The gp Calculator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 The general gp input line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 The PARI types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 GP operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.5 Variables and symbolic expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6 Variables and Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.7 User defined functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.8 Member functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.9 Strings and Keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.10 Errors and error recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.11 Interfacing GP with other languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.12 Defaults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.13 Simple metacommands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.14 The preferences file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.15 Using readline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
2.16 GNU Emacs and PariEmacs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Chapter 3: Functions and Operations Available in PARI and GP . . . . . . . . . . 63
3.1 Standard monadic or dyadic operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2 Conversions and similar elementary functions or commands . . . . . . . . . . . . . . . 71
3.3 Transcendental functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.4 Arithmetic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.5 Elliptic curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
3.6 L-functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
3.7 Modular symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
3.8 General number fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
3.9 Associative and central simple algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 288
3.10 Polynomials and power series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310
3.11 Vectors, matrices, linear algebra and sets . . . . . . . . . . . . . . . . . . . . . . . . . 326
3.12 Sums, products, integrals and similar functions . . . . . . . . . . . . . . . . . . . . . . 357
3.13 Plotting functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378
3.14 Programming in GP: control statements . . . . . . . . . . . . . . . . . . . . . . . . . 384
3.15 Programming in GP: other specific functions . . . . . . . . . . . . . . . . . . . . . . . 394
3.16 Parallel programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
3.17 GP defaults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
Appendix A: Installation Guide for the UNIX Versions . . . . . . . . . . . . . . . . 429
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

4

Chapter 1:
Overview of the PARI system
1.1 Introduction.
PARI/GP is a specialized computer algebra system, primarily aimed at number theorists, but has
been put to good use in many other different fields, from topology or numerical analysis to physics.
Although quite an amount of symbolic manipulation is possible, PARI does badly compared
to systems like Axiom, Magma, Maple, Mathematica, Maxima, or Reduce on such tasks, e.g. mul-
tivariate polynomials, formal integration, etc. On the other hand, the three main advantages of
the system are its speed, the possibility of using directly data types which are familiar to mathe-
maticians, and its extensive algebraic number theory module (from the above-mentioned systems,
only Magma provides similar features).
Non-mathematical strong points include the possibility to program either in high-level scripting
languages or with the PARI library, a mature system (development started in the mid eighties) that
was used to conduct and disseminate original mathematical research, while building a large user
community, linked by helpful mailing lists and a tradition of great user support from the developers.
And, of course, PARI/GP is Free Software, covered by the GNU General Public License, either
version 2 of the License or (at your option) any later version.
PARI is used in three different ways:
1) as a library libpari, which can be called from an upper-level language application, for
instance written in ANSI C or C++;
2) as a sophisticated programmable calculator, named gp, whose language GP contains most
of the control instructions of a standard language like C;
3) the compiler gp2c translates GP code to C, and loads it into the gp interpreter. A
typical script compiled by gp2c runs 3 to 10 times faster. The generated C code can be edited and
optimized by hand. It may also be used as a tutorial to libpari programming.
The present Chapter 1 gives an overview of the PARI/GP system; gp2c is distributed separately
and comes with its own manual. Chapter 2 describes the GP programming language and the gp
calculator. Chapter 3 describes all routines available in the calculator. Programming in library
mode is explained in Chapters 4 and 5 in a separate booklet: User’s Guide to the PARI library
(libpari.dvi.
A tutorial for gp is provided in the standard distribution: A tutorial for PARI/GP (tuto-
rial.dvi) and you should read this first. You can then start over and read the more boring stuff
which lies ahead. You can have a quick idea of what is available by looking at the gp reference card
(refcard.dvi or refcard.ps). In case of need, you can refer to the complete function description
in Chapter 3.
5
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