The equations seen on the classroom chalk boards are actual equations written by the real life John Nash.

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.

The film was shot in sequence in order to help Russell Crowe develop a consistently progressive manner of behavior.

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 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.

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.

Jennifer Connelly, who plays Alicia Nash, and Paul Bettany, who plays Charles, married shortly after the movie was released.

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."

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."

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).

One of John Nash's sons plays the orderly on the right, in the scene where Nash is being dragged down the hall.

John Nash is shown smoking in the film. In reality, he was a militant anti-smoker.

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.

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. In real life, Nash's delusions were only ever auditory, he has said he never suffered from visual delusions. This change was likely made for a cinematic approach.

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.

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.

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 film was shot 90% chronologically.

Salma Hayek was originally considered for the part of Alicia Larde because Alicia Nash is from El Salvador.

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.

An earlier trivia item describes John Nash as an anti-smoker. The movie is indeed accurate. He took up smoking in the '70s. This is mentioned early on in Chapter 45 of the book on which the movie is based.

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.

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.

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.

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.

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.

According to director Ron Howard, fake snow is used in some scenes. Although it was cold enough to snow, it would not accumulate.

A love scene with Russell Crowe and Jennifer Connelly was cut from the film.

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.

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.

Jennifer Connelly beat out Brittany Murphy for the role of Alicia Nash.

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.

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)).

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.

Although Russell Crowe did not want to meet the real John Nash for fear it might color his interpretation, he had little option when Nash showed up on the film set one day.

Rachel Weisz was offered the role of Alicia Nash and turned it down. She later played a famous mathematician herself in Agora (2009).

The Harvard scene is actually filmed at Manhattan College.

Bruce Willis, Kevin Costner, John Travolta, Tom Cruise, John Cusack, Charlie Sheen, Robert Downey Jr., Nicolas Cage, Johnny Depp, Ralph Fiennes, Jared Leto, Brad Pitt, Alec Baldwin, Mel Gibson, Sean Penn, Guy Pearce, Matthew Broderick, Gary Oldman and Keanu Reeves were all considered to star as John Nash.

For the third successive year, Russell Crowe was nominated for the Best Actor Oscar for his performance in a film which was also nominated for Best Picture, after The Insider (1999) and Gladiator (2000). He won Best Actor for Gladiator, while that film and this both won Best Picture.

For his performance, Russell Crowe won Best Actor at all the major precursor awards (Golden Globes, Screen Actors Guild, Critics Choice and BAFTA) but lost the Academy Award to Denzel Washington for his performance in Training Day (2001).

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.

Charlize Theron tried out for the role of Alicia Nash.

Robert Redford was considered as director, but withdrew due to scheduling conflicts.

Jennifer Connelly found the film so demanding that she briefly took up smoking.

This is one of Ron Howard's only films to feature an end title sequence. His other films all have opening credits.

Tom Cruise was considered to play John Nash.

The scene where Nash is sitting on the lawn talking to the students was filmed at the Bronx Community College campus (formerly NYU University Heights). If you look in the background you can see the Gould Memorial Library and behind that you can see the busts in the Hall of Fame of Great Americans.

When Nash works at MIT his office is room 101, like the infamous torture room of the Ministry of Love in George Orwell's 1984.

The infant used to portray their "son" in the dangerous bathtub scene was a female.

As of 2018, Jennifer Connelly won Best Supporting Actress for this film from her only Academy Award nomination.

Patrick Dempsey and Josh Lucas tested for the role of Martin Hensen.

Famke Janssen and Ashley Judd campaigned for the role of Alicia Nash.

Emily Watson was considered for the role of Alicia Nash, but was considered too British.

Russell Crowe knew the minute he read Akiva Goldsman's screenplay that he wanted to play John Nash.

During production, Ron Howard and Russell Crowe appeared on a video-only webcam stationed at the set's craft services area and also communicated with Steven Spielberg and Tom Cruise on the set of Minority Report (2002) in Los Angeles.

Mary McCormack was reportedly one of the finalists for the part of Alicia Nash, according to Ron Howard.

The producers had not originally thought of Jennifer Connelly for the role of Alicia. Portia de Rossi, Claire Forlani, Catherine McCormack, Meg Ryan, Rachel Griffiths and Amanda Peet were among the many actresses who lobbied for the role of Alicia.

Akiva Goldsman was inspired to write the screenplay after stumbling across a Vanity Fair article on John Nash. Intrigued, he started delving further into the man's history. Goldman is uniquely qualified to tackle such a subject as both of his parents were psychologists.

Mira Sorvino was considered for the role of Alicia Nash, but was considered too old.

The movie was named as one of "The 20 Most Overrated Movies Of All Time" by Premiere.

Anthony Rapp and Adam Goldberg also played friends in Dazed and Confused (1993).

Geena Davis, Teri Hatcher, Jennifer Jason Leigh, Mía Maestro, Rhona Mitra and Hilary Swank were considered for the role of Alicia Nash.

Julia Ormond auditioned for the role of Alicia Nash.

The scene where Russell Crowe and Jennifer Connelly are at a fancy party was filmed at the NJ University - Fairleigh Dickinson University. The campus, located in Madison NJ is a historical landmark - being the summer home of the Vanderbilt family.

Three separate trips were made to the Princeton University campus.

There are a few actors in this movie who acted in comic book films. Jennifer Connelly and Josh Lucas played Betty Ross and Talbot in Hulk (2003). Paul Bettany voiced JARVIS character in Iron Man (2008), Iron Man 2 (2010), The Avengers (2012), Iron Man 3 (2013) and Avengers: Age of Ultron (2015). Paul Bettany also played VISION character in Avengers: Age of Ultron (2015), Captain America: Civil War (2016) and future MCU movies. Russell Crowe played Jor-El in Man of Steel (2013). There is a cameo of Bryce Dallas Howard as Harvard Student for few seconds. She also acted in Marvel comic character Gwen Stacy in Spider-Man 3 (2007). Jennifer Connelly also plays Karen in Spider-Man: Homecoming (2017).

Seth Gabel who is married to Ron Howard's daughter, Bryce, appeared uncredited as a Harvard Student with Bryce when Nash is being driven away.

One of two Best Picture Oscar winners whose title begins with the word "A" (the other is A Man for All Seasons (1966).

Bryce Dallas Howard: The director's daughter and actress Bryce Dallas Howard appears 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.

In one scene, the imaginary roommate's niece runs through a field of birds. None of the birds move, proving she doesn't exist.

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.

John Nash wasn't taking medication when he won the Nobel. When he is speaking to the man about whether or not he will embarrass himself at the award ceremony he mentions he is taking new medication but in fact he stopped taking his meds and was using pure mind over matter to handle his disease. The screen writers mother was a psychiatrist and mentioned that it might promote others with mental illness to stop taking their medications so that's why Nash says that in the movie.

John Nash's wife played by Jennifer Connelly (Alicia Nash) and his imaginary friend played by Paul Bettany (Charles Herman) get married in 2003, 2 years after the 2001 release of "A Beautiful Mind".

Jennifer Connelly's Alicia Nash doesn't appear until around thirty minutes into the film.