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Dr. Ellie Arroway (Jodie Foster) works for the Search for Extraterrestrial Intelligence (SETI) program at the Arecibo Observatory in Puerto Rico. Fascinated by science and communication since she was a child, she listens to radio emissions from space hoping to find evidence of alien life. Science Advisor to the President David Drumlin (Tom Skerritt) pulls the funding from SETI because he believes the endeavor is futile. Arroway gains backing from secretive billionaire industrialist S. R. Hadden (John Hurt), which allows her to continue the project at the Very Large Array (VLA) in New Mexico.
Four years later, with Drumlin seeking to close SETI, Arroway discovers a signal repeating a sequence of prime numbers, apparently sent from the star system Vega some 26 light-years away. This announcement causes Drumlin and the National Security Council led by Michael Kitz (James Woods), to attempt to take control of the facility. Arroway's team then discover a video picture buried in the signal: Adolf Hitler's opening address at the 1936 Summer Olympics in Berlin. Arroway and her team postulate that this would have been the first television signal strong enough to leave Earth's atmosphere, taking 26 years to reach Vega and then transmitted back from there.
The project is put under tight security and its progress followed worldwide. Arroway learns that the signal also contains more than 60,000 pages of indecipherable data. The reclusive Hadden secretly meets with Arroway to provide the means to decode the pages, found when they are arranged in three dimensions rather than two-dimensional pages. The pages reveal schematics for a complex machine which is determined to be some kind of transport for a single occupant.
The nations of the world fund the construction of the machine at Cape Canaveral. An international panel is assembled to choose a candidate to travel in the machine. Although Arroway is a frontrunner to go, her hopes are scuppered by Christian philosopher Palmer Joss (Matthew McConaughey), a panel member whom Arroway met in Puerto Rico and had a brief romantic encounter. When he brings attention to her atheism, the panel selects Drumlin instead on the belief he would be more representative of humanity. However, on the day the machine is tested, a religious fanatic (Jake Busey) destroys the machine in a suicide bombing, killing Drumlin and many others.
A cancer-stricken Hadden, now in residence on the Mir space station, reveals to Arroway that a second machine was secretly made in Japan, and that Arroway will be the one to go. Outfitted with several recording devices, Arroway enters the machine's pod which is then dropped into four rapidly spinning rings causing the pod to apparently travel through a series of wormholes. Arroway sees a radio array-like structure at Vega and signs of an advanced civilization on another planet. She then finds herself in a surreal beachfront landscape, similar to a childhood picture she drew of Florida, and a blurry figure approaches that becomes her deceased father (David Morse). Arroway recognizes him as an alien taking her father's form and attempts to ask questions. The alien tells her that the familiar landscape and form were used to make their first contact easier for her and that this journey was just humanity's first step to joining other spacefaring species.
Arroway falls unconscious as she begins traveling back through a wormhole. She awakens to find herself on the floor of the pod, the mission control team repeatedly hailing her. She learns that, from outside the machine, it appears the pod merely dropped through the machine's rings and landed in a safety net. Arroway insists that she was gone for approximately 18 hours, but her recording devices show only static. A Congressional Committee is formed and speculates that the Vega signal and machine were a hoax designed by the now deceased Hadden. Although she cannot prove it happened, Arroway asks the committee to accept the truth of her testimony on faith. In a private conversation, Kitz and White House official Rachel Constantine (Angela Bassett) reflect on confidential information that, although Arroway's recording device only recorded static, it recorded 18 hours of it. Arroway and Joss reunite, and Arroway receives ongoing financial support to expand the SETI program.

Question: What does Arroway discover in a signal?
Answer: Repeating prime numbers