|Kenji is a part-time computer programmer from a poor family who has never had a girlfriend. Aside from the fact that he was almost selected to represent Japan in the Mathematics Olympiad he considers himself to be a complete loser. But, when Natsuki, the most popular girl in school, invites him to her grandmother's palatial home to pretend that he's her boyfriend, he gets caught up in an adventure that shakes the world.
A few quick mathematical references include his use of "modular arithmetic" to determine the day of the week Natsuki was born and a paper he is reading on Shor's factorization algorithm, but the primary piece of mathematics as far as the plot is concerned is Kenji's ability to break the encryption keys of OZ, a Facebook-like virtual world that is being taken over by a malevolent artificial intelligence named Love Machine.
The animation, both of the real world of Japan and the fantastical avatars of OZ, is quite beautiful and the story is compelling.
I was pretty impressed by some of the math that they put in the film. While the "breaking" of the 2056 digit integer seems rather mathematically questionable, I was pleasantly surprised to see a minor easter egg which I don't think anyone has pointed out in my quick Google check. In a sequence in the film at around 46:40, there is a computer readout of a list of people who had supposedly broken the aforementioned integer, among which includes some misspellings of famous mathematicians of past and present. See some screenshots here.
Some of the many highlights in the list include "Terence Taom", "Grigory Yakovlevich Pekelman", and "Stephen Volfram". Interestingly, it also includes "Paul Erdees" and "Godfrey Harold Hardy", who weren't alive at the time of the movie's release in 2009. I suppose they would be amused knowing that part of their legacy would be to cameo extremely briefly in a Japanese animated film made decades after they died.
The final decoded message also includes a quote by Socrates and "the magic words are squeamish ossifrage," the latter of which was also the real life solution to the RSA-129 challenge set out by Martin Gardner in his Scientific American column. While certainly not enough mathematical content to warrant anything higher than a 1 rating, it was an unexpected nod that was a nice tiny addition to the movie.
Stephen, thank you for these interesting observations. It did not occur to me to look that closely. But, looking over the list of names in the screenshots now I see misspellings of the names of many other familiar names including Gel'fand, Kontsevich, Gowers, Dijkgraf, Borcherds, Faltings, (mathematical fiction author) Rudy Rucker, ... in fact I see the inspiration for so many of the names on the list I am guessing that even the ones I do not recognize are based on the names of real people with connections to mathematics.