Trivia
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).
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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.
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One of John Nash's sons plays the orderly on the right in the scene where Nash is being dragged down the hall.
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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.
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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.
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John Nash visited the set, and Russell Crowe said later that he had been fascinated by the way he moved his hands, and he had tried to do the same thing in the movie.
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Salma Hayek was originally considered for the part of Alicia Larde because Alicia Nash is from El Salvador.
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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).
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Producer Brian Grazer won the rights to the project after John Nash and Alicia Nash chose him over competitor Scott Rudin; the Nashes had long resisted having a film made of their story.
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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.
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Barnard College professor Dave Bayer served as the math advisor on the film, and also was Russell Crowe's hand double for the scenes where he is writing equations on windows, etc.
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The film was shot in sequence in order to help Russell Crowe develop a consistently progressing manner of behavior.
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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.
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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.
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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.
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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.
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When John Nash is shown in the car riding from The Pentagon to Wheeler Labs, the voice on the radio is that of Sen. Joseph McCarthy, the infamous Communist hunter.
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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.
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According to director Ron Howard, fake snow is used in some scenes. Although it was cold enough to snow, it would not accumulate.
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Jennifer Connelly, who plays Alicia Nash, married Paul Bettany, who plays Charles, shortly after the movie was released.
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The equations seen on the classroom chalk boards are actual equations written by the real life John Nash.
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Cameo
Bryce Dallas Howard:
midway through the movie, after the math lecture when John Nash (Russell Crowe) is sedated by Dr. Rosen (Christopher Plummer), looking into the car as it drives away.
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Spoilers
In one scene, the imaginary roommate's niece runs through a field of birds. None of the birds move, proving she doesn't exist.
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