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I was looking for free alternatives to Magma to perform number theory computations, such as modular, factorization, etc. and I've found GAP and Pari.

What's the difference between them? Any other advise instead?

regards

New: Sage has a lot of features but it has two drawbacks: It's made with many different applications, great applications but independent, with few features, not properly merged on a single one. Each application has its commands, format...

And it's OK for Linux users but for Windows users has to be run virtualized, much slower.

I've found that the difference is that SAGE is better for abstract algebra problems and PARI for big number arithmetic.

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    I've grown fond of SAGE (www.sagemath.org). It's open source, free and has a lot easier syntax than many of the other math programs.2012-12-17
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    I'd also recommend sage... especially since it incorporates both GAP and Pari.2012-12-17
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    I personally prefere GAP. It has large libraries for group theory. But for number theory, SAGE might be your friend.2012-12-17
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    Maybe a bit out of topic here, but GAP actually has an interface to PARI/GP in the [Alnuth package](http://www.gap-system.org/Packages/alnuth.html)2013-04-22
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    Pari vs Gap: Pari is much faster for computationally hard number theory, but it is relatively easy to misuse (most pari objects are literally lists of numbers; in GAP objects are lists of numbers wrapped up in sugary coating so that you don't send the wrong kind of list of numbers to a function).2013-04-23

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