Software

Aliquot
A DOS tool to compute aliquot chains with small starting numbers, by Ivo Düntsch.
http://archives.math.utk.edu/software/msdos/dynamics/aliquot/.html

apfloat
High-performance arbitrary precision arithmetic package for C++ and Java.
http://www.apfloat.org/

ARIBAS
An interactive interpreter for big integer and multi-precision floating point arithmetic with a Pascal/Modula like syntax. It has several builtin functions for algorithmic number theory.
http://www.mathematik.uni-muenchen.de/~forster/sw/aribas.html

CALC
A calculator program for doing arbitrary precision integer arithmetic, written in ANSI C and Yacc, with a number of built-in functions for number theory, by Keith Matthews.
http://www.numbertheory.org/calc/krm_calc.html

cBigNumber
Portable C++ class for integers with unlimited range, designed to be fast, functional and simple in use.
http://www.imach.uran.ru/cbignum/

CLN - Class Library for Numbers
C++ library by Bruno Haible.
http://www.ginac.de/CLN/

ECMNET
Elliptic curve factoring software.
http://www.loria.fr/~zimmerma/records/ecmnet.html

Encyclopedia of Integer Sequences Seeker
EIS-Seeker is a libguile-based scheme interpreter devoted to the study of the Encyclopedia of Integer Sequences database. It is intended to help find new relations between sequences. C/Scheme, Posix.
http://sourceforge.net/projects/eisseeker

Factorizer
A Windows program to find all factors of any positive integer less than 2^31 - 1; decompose numbers into their prime constituents; find prime numbersand pairs of primes; and to plot Erdos-Kac and Palmen Color histograms. Down load free demo, order full version online.
http://www.hermetic.ch/factors/factors.htm

GIMPS FAQ list
About the Mersenne Prime Freeware.
http://www.mersenne.org/faq.htm

GNUBC
Supports arbitrary precision integer arithmetic calculations with interactive execution: it is a superior version of the basic bc program which comes with the UNIX operating system. Keith Matthews has written several number-theoretic programs.
http://www.numbertheory.org/gnubc/gnubc.html

Implementations of Extended LLL
By Wilberd van der Kallen.
http://www.math.uu.nl/people/vdkallen/lllimplementations.html

Java PHASER Project
An application for numerically and graphically analyzing dynamical systems.
http://www.math.miami.edu/~phaser

KANT / KASH
KANT is a software package for sophisticated computations in number fields and in global function fields. With KASH you are able to use the powerful KANT V4 functions within a shell.
http://www.math.tu-berlin.de/~kant/kash.html

Kaprekar Series Generator
A java application for determining the Kaprekar Constants and Series for numbers in various radixes.
http://kaprekar.sourceforge.net/

L
A C++ class library, datasets and command line program for computing zeros and values of L-functions, by Michael Rubinstein.
http://pmmac03.math.uwaterloo.ca/~mrubinst/L_function_public/L.html

Large Number Arithmetic in BASIC
Library for large-integer arithmetic, plus some number theory modules.
http://largeint.sourceforge.net/

LiDIA
C++ Library For Computational Number Theory.
http://www.cdc.informatik.tu-darmstadt.de/TI/LiDIA/

Magma Computational Algebra System Home Page
A large, well-supported software package for computationally hard problems in algebra, number theory, geometry and combinatorics.
http://magma.maths.usyd.edu.au/

MAGMA/PARI Calculator
Web interface for short programs in either language.
http://modular.fas.harvard.edu/calc/

Maple Worksheets by John Cosgrave
A selection of Maple (V.5) worksheets for teaching purposes.
http://www.spd.dcu.ie/johnbcos/Maple.htm

Math Archives Software
Mac/Number Theory section.
http://archives.math.utk.edu/software/mac/numberTheory/.directory.html

Math Archives Software
PC/Number Theory section.
http://archives.math.utk.edu/software/msdos/number.theory

MIRACL
Multiprecision Integer and Rational Arithmetic C/C++ Library.
http://indigo.ie/~mscott/

Multiprecision Software Directory
Fortran multiprecision software by David H. Bailey of Lawrence Berkeley Lab.
http://crd.lbl.gov/~dhbailey/mpdist/mpdist.html

NTL: A Library for doing Number Theory
A high-performance, portable C++ library providing data structures and algorithms for manipulating signed, arbitrary length integers, and for vectors, matrices, and polynomials over the integers and over finite fields.
http://www.shoup.net/ntl/

Number Field Sieve NET
A client/server system for NFSNET by Henrik Olsen.
http://www.iaeste.dk/~henrik/projects/nfsnet.html

Number-Theoretic Hacks
Algorithms implemented by David Eppstein.
http://www.ics.uci.edu/~eppstein/numth/

NZMATH
Python based number theory oriented calculation system, developed at Tokyo Metropolitan University. Downloads, mailing list.
http://tnt.math.metro-u.ac.jp/nzmath/

p-adic Arithmetic
A Mathematica package by Stany De Smedt.
http://library.wolfram.com/infocenter/MathSource/556/

Paul's TI-92 Number Theory Archive
Download site for number theory programs adapted from UBASIC/MALM for the TI-92.
http://www.geocities.com/Hollywood/2979/ntheory.html

SAGE
System for Arithmetic Geometry Experimentation: a framework for number theory, algebra, and geometry, initially designed for computing with elliptic curves and modular forms, by William Stein. Open source under GPL.
http://modular.fas.harvard.edu/SAGE/

Schulenberg and Associates
Factoring programs. Windows, free download.
http://www.schulenberg.com/

SIMATH
Computer algebra system for number theoretic applications.
http://tnt.math.metro-u.ac.jp/simath/

ZEN
A toolbox for computation in finite rings, by Florent Chabaud and Reynald Lercier. ANSI-C source, free download, GPL.
http://fchabaud.free.fr/English/ZEN/