Both the Sage notebook and codenode are similar projects that support development of Python programs via a web browser interface. They have been around for about 4 years, and full source code is available for both in case you want to setup your own server (there are dozens of Sage notebook servers used at universities around the world).

Microsoft has given significant funding to support the development of Sage and R. In the case of Sage the funding has always been "no strings attached". (I am director of the Sage project, and Sage is licenced under the GPL.)

I am a mathematics professor at the University of Washington in Seattle, and I published a textbook that I use in a course I teach. According to Washington State law, any royalties I receive as a result of purchases of my textbook by students in the course must be donated to the university (I tracked student purchases and donated a corresponding amount to UW). Second, I got permission from the publisher (Springer-Verlag) to make a free PDF version of the book available.

(1/10)^n for integer n is irrational in base 2 and the truncation was unavoidable.

Whether or not a number is irrational does not depend on the base. The number (1/10)^n is rational in any base. By irrational, maybe you meant "finite decimal expansion"?

Unrelated: The article starts with the example 599999999999999 - 599999999999998 = 0 in Google. Fortunately some software gives the correct result by default.

That's a good explanation. I have to emphasize though, that they actually found all the congruent numbers up to a trillion only under the completely unproven hypothesis that the Birch and Swinnerton-Dyer conjecture is true. It's entirely possible that this conjecture is false, and some of the numbers they found are actually not congruent numbers. However, part of the conjecture is known (by work of Coates and Wiles -- the same Wiles who proved Fermat's Last Theorem), so we do know that all numbers they didn't list are definitely not congruent numbers.

I own the 128GB RAM, etc., computer that the second group did the computation on. I have a Sun X4550 24TB disk array (ZFS) connected to it, but I only allocated a few terabytes of space for a scratch disk. They were well into the calculation when I found out what they were up to (I was initially annoyed, since they were saturating the network). I think they were just being polite to me and the other users by not using a lot more disk.

I asked them before this came out, and they said they didn't want to post their code on the press release in order to avoid being slashdotted. Seriously. I think the code is certainly available upon request, and will be made available later when the hoopla dies down. Much of it is in FLINT, which is part of Sage.

It is an *open problem* to show that there exists algorithm at all to decide whether a given integer N is a congruent number. Full stop. It's not a question of speed, or even skipping previous integers. We simply don't even know that it is possible to decide whether or not integers are congruent numbers. However, if the Birch and Swinnerton-Dyer conjecture is true (which we don't know), then there is an algorithm.

It is impacted. There are also quantum algorithms to break elliptic curve cryptography.
See, e.g., http://arxiv.org/abs/quant-ph/0301141

Microsoft also recently donated $32,000 to support the Sage Mathematics Software Project, which produced GPL'd free software. See the financial contributors list.

  -- William (Sage developer)