The student's "bad" tie shows stars and double helices. Although someone might have drawn those patterns already before the late 1940s when this scene takes place, the double helix as scientific symbol had its breakthrough not until 1953.
In the first scene in his dorm room at Princeton in 1947, Nash listens to a recording by the early-music ensemble Gothic Voices featuring soprano Emma Kirkby. The recording, of Hildegard von Bingen's "Columba aspexit," was not made until 1981; in fact, Emma Kirkby was not even born until 1949.
In 1994 when Nash is talking to the man about the Nobel Prize nomination and they are drinking tea, yellow packets of Splenda are clearly visible on the table. Splenda was not available in the U.S. until 1998.
There is a table telephone cord of the curly, coiled type. This was invented in 1954 but only appeared in 1956 . In the film the cord appears in 1953. The inventor is unknown but is thought to be an employee of Western Electric.
In the Pentagon scene, where Nash discovers the codes are map coordinates, one of the coordinates he reads off is "67-46-90", presumably in degrees, minutes and seconds. But in the degree-minute-second coordinate system, seconds go up only to 59, not 90.
Contrary to what the Nobel representative claims, "the Sveriges Riksbank prize in economic sciences in memory of Alfred Nobel" is not privately funded. It is financed by the Swedish state bank. (The five "genuine" Nobel prizes are funded by the legacy of Alfred Nobel.)
As Nash walks on the Harvard campus on his way to the math conference (shortly before being chased by Rosen and his men), he drops his bag behind him to hug his roommate and the young girl. During his conversations with them, the bag remains on the ground. Within one second, just before Nash walks off, the bag goes from being on the ground to being clutched to his chest.
In a later scene, when Nash's former rival and now President of Princeton are walking and said President offers Nash a job in the Spring, we see that Nash has aged quite a bit but the friend has not, with regard to earlier scenes.
When Nash and Parcher are walking to the warehouse the wide shot so well lit shows the right side of the stairway they pass as lit from behind the left wall. On the tighter shot when they actually cross in front of that stairway there is no light on the wall at all. Then in the wide shot again the light is there.
When Alicia calls Dr. Rosen about John's breakdown, he makes a house-call to their house. He was a psychiatrist at MacArthur Hospital (McLean Hospital in Belmont, MA), but drove all the way to Princeton, NJ where Alicia and John were living.
Several scenes with Nash sitting on the lawn are clearly filmed in the old NYU University Heights Campus (now Bronx Community College). If you look in the background you can see the busts in the "Hall of Fame of Great Americans".
Strictly speaking, John Nash didn't win the Nobel Prize because there isn't a prize for Economics or Mathematics. (Alfred Nobel who willed his estate to the Nobel foundation saw no need for a prize in mathematics.) In 1969 the Swedish Central Bank established the "The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel". This prize is presented in the same ceremony and is therefore often mistaken for a proper Nobel Prize. It is even often referred to as the "Nobel Prize in Economics" in daily conversation; the fictional character of President Jed Bartlet on The West Wing was also presented as a Nobel Prize winner (for economics) with the show also not making the real-world distinctions.
John Nash didn't receive the Nobel prize alone, but with colleague Reinhard Selten and Hungarian-born János Harsányi. "Game Theory" was initiated by Hungarian-born John von Neumann and Austrian-born Oskar Morgenstern in 1944.
John's and Alicia's baby son first appears in 1955 depicted as being several months old. Then in 1956, he is played by the same baby and appears not to have aged a day while in fact he should have been depicted as a toddler. In reality, the son wasn't even born until 1959.
John and Alicia Nash did not live in Princeton, NJ, despite what the end title card said. In reality, they lived in Princeton Junction, NJ, within walking distance of the Princeton Junction train station.
After Helinger tells John he will not receive any placement, the camera pulls back on John as he stands in the doorway. The actor's 'T' mark can be seen on the floor to the far right of the screen where John had previously been standing.
The goof item below may give away important plot points.
When Nash arrives at MIT after having come from the pentagon he is wearing a topcoat though everyone complains about the air-conditioning is broken and it followed by a scene in which opening windows because of the heat facilitates the film's "meet cute". However this might not be a gratuitous error but a foreshadowing of Nash's mental disturbances.
When Alicia goes to the abandoned house where John has been dropping off envelopes, she waits until the gate is fully open before entering the courtyard. In the next shot, the gate is about 3/4 open and still opening as she enters.