The play concerns Catherine, the daughter of Robert, a recently deceased mathematical genius in his fifties and professor at the University of Chicago, and her struggle with mathematical genius and mental illness. Catherine had cared for her father through a lengthy mental illness. Upon Robert's death, his ex-graduate student Hal discovers a paradigm-shifting proof about prime numbers in Robert's office. The title refers both to that proof and to the play's central question: Can Catherine prove the proof's authorship? Along with demonstrating the proof's authenticity, the daughter also finds herself in a relationship with 28-year-old Hal. Throughout, the play explores Catherine's fear of following in her father's footsteps, both mathematically and mentally and her desperate attempts to stay in control.
The play opens with Catherine sitting alone in the backyard of her large, old house. Robert, her father, approaches her with a bottle of champagne to celebrate her 25th birthday. Catherine complains that she hasn't done any worthwhile work in the field of mathematics, at least not to the same level as her father, a well-known math genius. He reassures her that she can still do good work as long as she stops laying in bed til all hours and wasting time reading magazines. Catherine confesses she's worried about inheriting Robert's inclination towards mental instability. He begins to comfort her but then alludes to a "bad sign" when he points out that he is, in fact, dead. He died a week ago. Robert disappears as Catherine dozes off. She awakens when Hal, one of Robert's students, exits the house. He's been studying the hundreds of notebooks Robert left behind after his death, looking for any work that could be published. Catherine assures him that the notebooks are filled with scribbles and nonsense since her father wrote them when he was at his most delusional. Hal, attempting to flirt invites her to go see his band later that night. Catherine becomes suspicious of him and demands to see what's in his backpack. She roots through it to find nothing but becomes infuriated when a notebook falls out of Hal's jacket. She dials the police while accusing him of trying to steal her father's work and pass it off as his own. He admits that he was sneaking it away but only to give it back to her later as a birthday present. He opens to a page that Robert wrote during a time when he was lucid. In it, Robert writes it's a "good day" and thanks to Catherine for taking care of him and expresses hope for the future. Hal leaves Catherine with the notebook. She begins to cry until she hears police sirens.
The next day, Claire, Catherine's sister who just flew in from New York is setting up a large brunch for them in the backyard. Catherine enters and Claire tries to goad her into idle chitchat as Catherine quietly seethes. Claire declares she's getting married and invites Catherine to stay with her and her fiancee in New York. Catherine assures her she'll come in February for the wedding, but Claire keeps pressing her to go earlier. When Catherine demands to know why Claire is inundating her with questions, Claire tells her the police came over earlier to check in on Catherine. Catherine admits to calling the police the previous night and tries to explain her altercation with Hal but only ends up sounding unhinged to the dubious Claire. Hal appears and asks to continue his work sorting the notebooks. Catherine lets him inside and Claire drops a hint for Catherine to try flirting with Hal by offering a bagel. Catherine storms into the house.
Later that night, after the funeral, Claire holds a party in the house for her friends as well as Hal and Robert's students. Catherine escapes to the porch where Hal finds her and offers her a beer. Hal confesses that he's not so sure about his own mathematical abilities since he considers math to be a "young man's game". Catherine tries to reassure him with a quote from Gauss. Hal responds by kissing her, much to Catherine's surprise. He apologizes for trying to steal the notebook and she apologizes for calling the police. They kiss again and Hal asks Catherine if she remembers meeting him years earlier. She says she does and recalls she thought he was "not boring". They continue to kiss.
The next morning Catherine sits outside. Hal exits the house and tells her he'd like to spend the rest of the day with her. Catherine gives him a key to Robert's desk and tells him to look inside. He goes into the house. A moment later, Claire comes into the backyard, extremely hungover. Catherine, now in a good mood, tries to make nice with Claire. Claire takes the opportunity to continue to push Catherine to moving to New York. Catherine asks why she would move to New York to which Claire confesses that she's selling the house. Catherine becomes enraged at the idea and she accuses Claire of abandoning her to take care of their sick father alone. Claire insists that the reason she did so was to keep working to pay for the house as well as Catherine's education. Catherine reveals that she had to quit school to tend to Robert and then accuses Claire of trying to have her committed. Claire admits that she's researched doctors and facilities for Catherine but insists that she wasn't planning on having her committed. In the middle of the row, Hal appears clutching a notebook, barely containing his excitement. He tells Claire that Catherine is in possession of one of Robert's notebooks which holds a very important proof. Claire asks Catherine where she found it and Catherine tells them she didn't find it. She wrote it.
We flashback to years earlier, with Robert sitting in the backyard. Catherine tells him she thinks he's getting better and he agrees. She blurts out that she's decided to go to college in a couple months, funded by Claire, but promises she'll be only a short drive away if he were to need her again. Robert protests and demands to know why she waited so long to tell him. When she points out that he hadn't been well until recently and was, at one point, trying to decode extraterrestrial messages in library books, he becomes upset. Hal interrupts, much to his embarrassment, to present his final dissertation to Robert. Robert assures him they'll work out the problem points together, then suddenly realizes he's forgotten Catherine's birthday. He apologizes and offers to take her out to dinner. Catherine invites Hal along but he says he can't go. Catherine shows Hal out and Robert sits down to write a notebook entry, declaring it to be a "good day".
We flash forward to where Act I left off. Catherine declares she was the one who wrote the proof and is met with incredulity by both Hal and her sister. The handwriting is very much like Robert's and Hal questions Catherine's mathematical abilities given that she only had a couple years education at Northwestern. Catherine tells him that her real education was living with Robert. When Hal offers to show it to other math experts to confirm the authenticity of the proof, Catherine refuses. She tells Hal she trusted him and then accuses him of having no talent and being past his prime. Hal storms off and Catherine begins to rip the notebook apart. Claire gets it away from her and Catherine runs into the house.
Later, Hal attempts to visit Catherine and apologize for his behavior. Claire stops him and tells him Catherine won't talk to her, let alone Hal. Claire accuses him of sleeping with Catherine despite her being unstable. Hal argues that he had no bad intentions and insists Catherine is stronger than Claire thinks. He requests to have the notebook to verify its authenticity with fellow mathematicians. Claire gives it to him and tells him she's taking Catherine with her to New York the next day. She expresses concern for Catherine's future mental stability.
We flashback to Robert in the backyard, sitting in the cold and writing furiously. Catherine enters and reprimands him for sitting in the cold with no jacket. Robert tells her it's too hot in the house and that the cold is better for helping him work. Catherine is shocked that he's working again and he assures her that he's sharper than ever. She's ecstatic that his previous mental instability has passed and asks to see his work. He says he'd love for her to take and look and asks if she'd like to take time off school to work with him. Before she decides, Robert insists she look at his latest idea and thrusts a notebook into her hands. Catherine glances at it and becomes quiet. She tells him they need to go inside and Robert explodes with fury. He yells at her to read what he's wrote. She reads aloud, a nonsensical, rambling paragraph about winter and books and the cold. It's obvious that Robert's mind is deteriorating as it had been before. Catherine begins to cry as Robert descends into confusion and begins to shiver uncontrollably. Catherine tries to take him inside when he asks her not to leave. She promises she won't.
We flash forward to Claire in the backyard. Catherine enters with her suitcase. She asks Claire about life in New York. Claire mentions potential schools or jobs for Catherine but Catherine is quick to mock her by making ridiculous demands for a Freudian psychiatrist who will listen as she blames all her problems on Claire. Claire begins to cry and throws Catherine's plane ticket in front of her before storming off. Hal enters and tells Catherine that the proof checks out and apologizes for not believing her. Catherine tells him there's no proof that she wrote it and he can claim it as his own if he wants. Hal tells her he believes she's the one who wrote it and offers to read through it with her. Catherine admits she knows she's like her father but is terrified of becoming like her father. Hal reassures her that maybe she'll be better. Catherine opens the proof and begins to talk through it with Hal.