The Pentagon office scene was filmed in the basement of Keating Hall on Fordham University's Bronx campus (the same room was used in the filming of the Georgetown University language lab scene in The Exorcist (1973)).
The Riemann Hypothesis mentioned throughout the movie is a real and famous problem in mathematics that has gone unsolved (it has not been proved yet) for nearly 150 years. Many other important theories have been proved on the condition that the Riemann Hypothesis holds, hence its importance. In the year 2000, the Clay Mathematics Institute of Cambridge, Massachusetts listed the Riemann Hypothesis as one of seven "Millennium Prize Problems" and offered a $1,000,000 reward to the person that proves it.
Nash's mutterings after he loses the board game (along the lines of "the game is flawed," "I had the first move, I should have won") are in reference to "Game Theory" - the economic theory that John Nash is probably most famous for.
While this film is inspired by the life of John Nash, there were elements from his life that were deliberately omitted: 1) he was married twice, both to the same woman (Alicia Nash); 2) in the past, he had several affairs with both men and women; 3) he was arrested by the police by scandal; 4) He fathered a child out-of-wedlock in his twenties; 5) he believed that through his mental illness the extra-terrestrials spoke him, giving his advanced knowledge by means of cosmic connection with them; 6) he tried to renounce to his American nationality some times, in the belief that the USA government pursued him; and 7) he made numerous anti-Semitic comments during his period of extreme mental illness, most of which equated Jews with world Communism.
To create the "golden" look of the campus scenes early in the film, the filmmakers took a low-contrast stock (Fuji F-400 8582) and exposed it to an orange light before loading it into the camera for shooting.
After coming up with the idea for his revolutionary paper, John Nash goes and shows a manuscript of it to Helinger (Judd Hirsch). The manuscript is an actual copy of the original article, published in the specialized journal "Econometrica", under the title "The Bargaining Problem". (Figure 1 of the original paper, appears in the manuscript shown in the movie).
The scene towards the end of the film where John Nash contemplates drinking tea is based on a true event when Russell Crowe met the real John Nash. He spent 15 minutes contemplating whether to drink tea or coffee.
According to a 2001 Entertainment Weekly article on this film, the filmmakers originally wanted to mention John Nash's homosexuality, but they feared the film would make the wrong connection between homosexuality and schizophrenia, so they abandoned it. This connection, according to the article, was based on several now-discredited psychological studies that first appeared in the late 1950s.
The problem that John Nash writes on the blackboard in his lecture is a real one (unlike in other movies, where math on boards is usually either too simple or fake). There is an important theorem in mathematical physics that directly says the answer to this is 1. Later, when he discusses the problem with Alicia Nash, he makes additional restrictions for the solution, without which the problem is much harder, so he is pretty confident she didn't solve it.
The Nobel Prize ceremony was filmed in Prudential Hall at the New Jersey Performing Arts Center (NJPAC) in Newark, NJ. The filming for that one scene, including set up, make up, etc, took over 8 hours. However, the scene in the lobby afterwards was filmed at another location.
Screenwriter Akiva Goldsman had plenty of personal experience to draw on in developing the story; he had previously worked as a child care counselor and had developed a method for training mental health workers, and had also grown up in a house where his parents had established a group home for emotionally disturbed children.
Nash is the co-inventor (independently from the Danish mathematician Piet Hein) of the strategy game Hex. This was caused by his frustration after he was defeated at Go, and he wanted to create a "perfect" game. The scenes where he does that were shot but then deleted from the final cut, and can be found in the DVD extras or online.