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.
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. He thought it would help him get into the character.
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.
John and Alicia Nash were killed in a car accident on May 23, 2015, when the driver of a taxi they were riding in lost control of the car and collided with a guard rail and a second car on the New Jersey Turnpike. The New York Times said that although the two drivers suffered only minor injures, the Nashes were "ejected from the cab and pronounced dead at the scene." John was 86 and Alicia was 82.
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 fifteen minutes contemplating whether to drink tea or coffee.
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).
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.
John 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.
Professor Peyton Young was interviewed on the BBC radio programme "More or Less". He had dinner with John and Alicia Nash the night the movie opened and asked John Nash what he thought of it. He said "I liked it a lot. But it wasn't me."
John Nash attended the Carnegie Institute of Technology on a full scholarship. One of his professors was so impressed with him that he wrote him a letter of recommendation for Princeton that contained only one line; "This man is a genius."
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.
Akiva Goldsman's creative take on the project was to avoid having viewers understand they are viewing an alternate reality until a specific point in the film. This was done to rob the viewers of their understanding, to mimic how Nash comprehended his experiences. Ron Howard agreed to direct the film based on the first draft. He asked Goldsman to emphasize the love story of Nash and his wife; she was critical to his being able to continue living at home.
During filming, Ron Howard decided that Nash's delusions should always be introduced first audibly and then visually. This provides a clue for the audience and establishes the delusions from Nash's point of view.
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.
Dave Bayer, a professor of Mathematics at Barnard College, Columbia University, was consulted on the mathematical equations that appear in the film. Bayer later said that he approached his consulting role as an actor when preparing equations, such as when Nash is forced to teach a calculus class, and arbitrarily places a complicated problem on the blackboard. Bayer focused on a character who did not want to teach ordinary details and was more concerned with what was interesting. Bayer received a cameo role in the film as a professor who lays his pen down for Nash in the pen ceremony near the end of the film.
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 filmmakers developed a technique to represent Nash's mental epiphanies. Mathematicians described to them such moments as a sense of "the smoke clearing", "flashes of light" and "everything coming together", so the filmmakers used a flash of light appearing over an object or person to signify Nash's creativity at work.
Three people closely associated with the movie unexpectedly passed away in vehicle accidents in 2015, all within one month. John Nash and Alicia Nash died together in a traffic accident on May 23; composer James Horner died in a plane crash on June 22.
Greg Cannom was chosen to create the makeup effects, specifically the age progression of the characters. Russell Crowe had previously worked with Cannom on The Insider (1999) and Ron Howard had worked with him on Cocoon (1985). Each character's stages of makeup were broken down by the number of years that would pass between levels. Cannom stressed subtlety between the stages, but worked toward the ultimate stage of "Older Nash". The production team originally decided that the makeup department would age Russell Crowe throughout the film; however, at Crowe's request, the makeup was used to push his look to resemble the facial features of John Nash. Cannom developed a new silicone-type makeup that could simulate skin and be used for overlapping applications; this shortened make-up application time from eight to four hours. Crowe was also fitted with a number of dentures to give him a slight overbite in the film.
A running discussion between Ron Howard and James Horner was the concept of high-level mathematics being less about numbers and solutions, and more akin to a kaleidoscope, in that the ideas evolve and change. After the first screening of the film, Horner told Howard: "I see changes occurring like fast-moving weather systems." He chose it as another theme to connect to Nash's ever-changing character.
After producer Brian Grazer first read an excerpt of Sylvia Nasar's book A Beautiful Mind in Vanity Fair magazine, he immediately purchased the rights to the film. He eventually brought the project to Ron Howard, who had scheduling conflicts and was forced to pass. Grazer later said that many A-list directors were calling with their point of view on the project. He eventually focused on a particular director, who coincidentally was available only when Howard was also available. Grazer chose Howard.
James Horner chose Charlotte Church to sing the soprano vocals after deciding that he needed a balance between a child and adult singing voice. He wanted a "purity, clarity and brightness of an instrument" but also a vibrato to maintain the humanity of the voice.
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.