PARI/GP

PARI/GP

PARI/GP in use onWindows XP
Developer(s)	Henri Cohen,Karim Belabas, et al., at theUniversité Bordeaux 1
Stable release	2.15.5 / 23 February 2024(23 February 2024)
Repository	pari.math.u-bordeaux.fr/git/pari.git
Written in	C
Operating system	Cross-platform
Type	Computer algebra system
License	GNU General Public License
Website	pari.math.u-bordeaux.fr

PARI/GPis acomputer algebra systemwith the main aim of facilitatingnumber theorycomputations. Versions 2.1.0 and higher are distributed under theGNU General Public License.It runs on most commonoperating systems.

System overview

The PARI/GP system is a package that is capable of doing formal computations on recursive types at high speed; it is primarily aimed at number theorists. Its three main strengths are its speed, the possibility of directly usingdata typesthat are familiar tomathematicians,and its extensive algebraic number theory module.

The PARI/GP system consists of the following standard components:

• PARIis aClibrary,allowing for fast computations, and which can be called from a high-level language application (for instance, written in C,C++,Pascal,Fortran,Perl,orPython).
• gpis an easy-to-use interactivecommand line interfacegiving access to the PARI functions. It functions as a sophisticated programmable calculator which contains most of the control instructions of a standard language like C.GPis the name ofgp's scripting language which can be used to programgp.

Also available isgp2c,the GP-to-C compiler, which compiles GP scripts into the C language and transparently loads the resulting functions intogp.The advantage of this is thatgp2c-compiled scripts will typically run three to four times faster.gp2cunderstands almost all of GP.

PARI/GP performsarbitrary precisioncalculations (e.g., thesignificandcan be millions of digits long—and billions of digits on64-bitmachines). It can computefactorizations,performelliptic curvecomputations and performalgebraic number theorycalculations. It also allows computations withmatrices,polynomials,power series,algebraic numbersand implements manyspecial functions.

PARI/GP comes with its own built-in graphicalplottingcapability. PARI/GP has some symbolic manipulation capability, e.g., multivariate polynomial and rational function handling. It also has some formalintegrationanddifferentiationcapabilities.

PARI/GP can be compiled withGMP(GNU Multiple Precision Arithmetic Library) providing faster computations than PARI/GP's native arbitrary-precision kernel.

History

PARI/GP's progenitor was a program namedIsabelle,an interpreter forhigher arithmetic,written in 1979 byHenri CohenandFrançois Dressat theUniversité Bordeaux 1.

PARI/GP was originally developed in 1985 by a team led by Henri Cohen at Laboratoire A2X and is now maintained byKarim Belabasat the Université Bordeaux 1 with the help of many volunteer contributors.

Etymology

The name PARI is a pun about the project's early stages when the authors started to implement a library for "Pascal ARIthmetic" in thePascal programming language(although they quickly switched toC), and after"pari de Pascal"(Pascal's Wager).

The first version of thegpcalculator was originally called GPC, for Great Programmable Calculator. The trailing C was eventually dropped.

Usage examples

Below are some samples of thegpcalculator usage:

?\p 212
realprecision = 221 significant digits (212 digits displayed)
? (1.378-0.09143*I)^(14.87+0.3721*I)
time = 0 ms.
%1 = 80.817082637557070449383034933010288336925078193546211741027496566803185
11092579265743992920628314516739962724446042667886245322716456966120413965187
3272488827365261487845201056199035423784093096984005713791800191 - 94.8384618
89186304973351271821601500916571303364865064205039706592481303045713982306764
33264430511752515705768858710051382035377195497482934017239179757538824688799
0680136241031895212412150770309289450962931402933*I

? 123456! + 0. \\ slower than gamma(123457) which uses floating point
time = 1,656 ms.
%2 = 2.6040699049291378729513930560926568818273270409503019584610185579952057
37967683415793560716617127908735520017061666000857261271456698589373086528293
4317244121152865814030204645985573419251305342231135573491050756 E574964

? sin(x)
time = 0 ms.
%3 = x - 1/6*x^3 + 1/120*x^5 - 1/5040*x^7 + 1/362880*x^9 - 1/39916800*x^11
+ 1/6227020800*x^13 - 1/1307674368000*x^15 + O(x^17)

? for(z=25,30, print (factor(2^z-1)))
[31, 1; 601, 1; 1801, 1]
[3, 1; 2731, 1; 8191, 1]
[7, 1; 73, 1; 262657, 1]
[3, 1; 5, 1; 29, 1; 43, 1; 113, 1; 127, 1]
[233, 1; 1103, 1; 2089, 1]
[3, 2; 7, 1; 11, 1; 31, 1; 151, 1; 331, 1]
time = 5 ms.

? K = bnfinit(x^2 + 23); K.cyc
time = 1ms.
%4 = [3]
/* This number field has class number 3. */

See also

• SageMath,a multiple-software mathematical package which includes PARI/GP as one of its components
• List of computer algebra systems

External links

• PARI/GP Development Headquarters
• PARI/GP - Mathematical software - swMATHwith a collection of references
• SIGSAM Computer Algebra Software
• Rosetta Code: PARI/GP(sample programs)
• Catalogue of GP/PARI Functions;also in downloadablegzippedtarballarchive:Stable Branch

•PARI/GP online calculator

• Port of PARI/GP to Android