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Last Updated on May 5, 2015, by eNotes Editorial. Word Count: 1271

Act 1, Scene 1
Proof begins at one o’clock in the morning on the porch of a house in Chicago. Catherine sits in a chair, exhausted, and is startled when she realizes her father, Robert, is there. Robert gives her a bottle of champagne and wishes her happy birthday. He wants her to celebrate her birthday with friends, but she says she has none. Robert expresses concern about her, saying that she sleeps until noon, eats junk food, and does not work. He tells her to stop moping. She has potential and there is still time. It transpires that Robert did his best work by the time he was in his mid-twenties. After that, he became mentally ill. Catherine is worried that she will inherit the illness.

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It then transpires that Robert died a week before, of heart failure, and the funeral is the next day.

Hal, a former student of Robert’s, enters. He has been working on Robert’s notebooks, but Catherine says there is nothing valuable in them. Hal invites her to hear him play in a rock band, but she is not interested. He speaks about how he admired her father, who helped him through a difficult period in his doctoral studies. This was four years ago, when Robert’s illness went into remission. Catherine, fearing that Hal may be taking one of her father’s notebooks from the house without permission, demands to see his backpack. She finds nothing there, but as he is about to leave, a notebook falls from his jacket pocket. She accuses him of stealing it and calls the police. He protests that in the notebook, Robert wrote something appreciative about Catherine on her birthday four years ago. Hal was going to wrap the notebook and give it to her.

Act 1, Scene 2
The next morning, Catherine and Claire, who has arrived from New York, are drinking coffee. Claire tries to be kind, but Catherine is not receptive. Claire quizzes Catherine about Hal and about why she called the police, but Catherine resents the questioning. Hal enters unexpectedly, and there is a moment of confusion as Catherine berates her sister. Hal quickly exits, leaving Claire saying that decisions must be made. She wants Catherine to stay with her in New York.

Act 1, Scene 3
That night, there is a party following the funeral. Catherine is on the porch when Hal, who has been playing in the band, approaches her. He compliments her on her dress and talks about how mathematicians consider they are past their peak after the age of twenty-three. He refers to them as men, but Catherine mentions Sophie Germain, an eighteenth-century Frenchwoman who did important work on prime numbers. Catherine apologizes for her behavior the day before, and Hal confides that he thinks his work in mathematics is trivial. They talk about how elegant Robert’s work was. Catherine then surprises Hal by kissing him. Hal reminds her of when they first met, four years ago, and they kiss again.

Act 1, Scene 4
Hal and Catherine have spent the night together, and the next morning she gives him a key to the bottom drawer of her father’s desk. Claire enters with a hangover. She tells Catherine that she would like her to move to New York. Catherine says she would prefer to stay in Chicago, but Claire replies that she has already sold the house. They quarrel. Catherine complains that Claire never helped to take care of their father; Claire replies that she worked fourteen-hour days so she could pay off the mortgage on the house. She says that Robert should have been sent to an institution, but Catherine disagrees. Hal returns with a notebook. Inside it, he says, is a proof of a theorem about prime numbers. If it checks out, it will show that when Robert was supposedly insane, he was doing some of the most important math work in the world. Catherine stuns him by saying that it was she who wrote it.

Act 2, Scene 1
It is a September afternoon four years earlier. Robert and Catherine talk on the porch. Catherine says she has enrolled as a math major at Northwestern. She tells him that since he has been well for nearly seven months, he does not need her there all the time. Robert is not happy about her decision and says she should have discussed it with him. Hal enters. At this time, he is Robert’s graduate student, and he brings a draft of his dissertation. Robert says he will look it over and tells Hal to come by his office in a week. Then, he realizes that it is Catherine’s birthday, and he had forgotten it. He is annoyed with himself, but Catherine tells him not to worry. They agree to go out to dinner. As Catherine goes out to dress, Robert begins writing in his notebook.

Act 2, Scene 2
Catherine, Hal, and Claire discuss the newly discovered notebook. Catherine insists that she wrote the proof, working on it for years after she dropped out of school. Hal and Claire are skeptical. Claire thinks the proof is written in her father’s handwriting. She suggests that Catherine talk them through it to convince them, but Hal says that would not prove anything, since her father might have written it and explained it to Catherine later. Catherine is unhappy that they do not believe her. She says she trusted Hal and wanted him to be the first person to see the proof. He still cannot believe that she wrote it, since to do so she would have to have been as good as her father. After Catherine snaps at him, he exits. Catherine and Claire struggle over the notebook and Catherine throws it to the floor.

Act 2, Scene 3
The next day, Claire berates Hal for taking advantage of Catherine and sleeping with her. She refuses to let Hal talk to her, but she does let him take the notebook. She tells Hal to figure out what is in there and advise the family about what to do.

Act 2, Scene 4
It is winter, three and a half years earlier. Robert is on the porch in the cold, writing in a notebook. When Catherine, who is a student at Northwestern, arrives, he tells her that he is working again. He feels he has regained all his intellectual brilliance and is excited about what he will be able to produce. He wants her to collaborate with him and hands her his notebook, which Catherine reads slowly. It is confused, rambling nonsense. She puts her arm around him and takes him inside the house.

Act 2, Scene 5
Back in the present, Claire and Catherine prepare to leave for New York. At first, they appear to be getting on well, but when Claire tells her how much she will love New York, Catherine gives sarcastic replies, and the two women quarrel. Claire exits, upset. Hal enters. He is excited. The proof has checked out. Catherine is not surprised and tells him to publish it. He now believes that it is her work because it uses new mathematical techniques that he thinks Robert would not have known. He wants Catherine to talk about her work so he can understand it better. Catherine is upset that he did not trust her in the first place. He hands her the book. She says that doing the proof was just a matter of connecting the dots. Her father knew nothing of her work. Hal asks her to go through it with him, and she picks up the book, finds a section, looks at him, and begins speaking.

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