In comments to my previous post, Rytis Jursenas pointed at an entire article in the Notices of AMS discussing several bugs in Mathematica. Since such problems are apparently quite common, I feel it deserves a separate post.
Here it comes:
"The Misfortunes of a Trio of Mathematicians Using Computer Algebra Systems. Can We Trust in Them?" by Antonio J. Durán, Mario Pérez, and Juan L. Varona
(just in case, here is the preprint)
"...We have been using Mathematica as a tool in our mathematical research. All our computations with Mathematica have been symbolic, involving only integers (large integers, about 10 thousand digits long) and polynomials (with degree 60 at most), so no numerical rounding or instability can arise in them, and we completely trusted the results generated by Mathematica. However, we have obtained completely erroneous results."
"...Software bugs should not prevent us from continuing this mutually beneficial
relationship in the future. However, for the time being, when dealing with a problem whose answer cannot be easily verified without a computer, it is highly advisable to perform the computations with at least two computer algebra systems."
Well, I didn't expect such a conclusion to be drawn in 2014.