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Proof invites audiences into the mind of Catherine to decipher between her intellectual brilliance and potential mental illness. Catherine had cared for her father, Robert, through a lengthy mental illness. Upon Robert’s death, one of his previous Ph.D. students, Hal, appears to analyze Robert’s remaining notes to determine if there is any important work that should be published. It is during this review that a paradigm shifting proof about prime numbers is discovered in Robert’s home office.