## Section: Application Domains

### Computational Number Theory Systems

We have strong ties with several computational number theory systems, and code written by members of the project-team can be found in the Magma, Pari/GP, and Sage software tools.

Magma(http://magma.maths.usyd.edu.au/magma/ ) is the leading computational number theory software. It also has some features of computer algebra (algebraic geometry, polynomial system solving) but not all of what is expected of a computer algebra system. It is developed by the team of John Cannon in Sydney.

Pari/GP(http://pari.math.u-bordeaux.fr ) is a computational number theory system which comes with a library which can be used to access Pari functions within a C program. It has originally been developed at the Bordeaux 1 University, and is currently maintained (and expanded) by Karim Belabas, from Bordeaux University. It is free (GPL) software. We sometimes use it for validation of our algorithms. Again, some code written by members of the project-team is incorporated into Pari.

Sage(http://sagemath.org ) is an open-source computer algebra system. Its development was initiated by William Stein (Univ. of Washington, Seattle). Instead of reinventing the wheel, Sage incorporates the most efficient open-source packages in each domain, for example Singular , Pari/GP, Ntl , Linbox , and the software tools Mpfr and Gmp-ecm developed by Cacao . Although quite new, there is already a strong community of active developers around Sage. This system is a good alternative to Maple, Mathematica, and Magma to better disseminate our research in the future.