This episode, airing in 1989, predicts that Fermat's Last Theorem would have gone unsolved for 800 years as of the mid-24th century. It was actually solved in 1993 by Princeton University Professor Andrew Wiles. In 2016, he won the Abel prize in recognition of his accomplishment.
Tracy Tormé wrote 'The Royale' as a story of a man trapped to relive his most pleasant memory until it becomes nightmarish. Tormé was dissatisfied with the final product, which bears little resemblance to his original concept, and used the pseudonym Keith Mills in the credits.
The Royale book begins, "It was a dark and stormy night." This opening line was first used in 1830 by Sir Edward George Bulwer-Lytton for the novel 'Paul Clifford' and has been imitated and spoofed by countless other authors, including W.W. Jacobs and Charles M. Schulz. It is often used to make fun of humorously incompetent or overly melodramatic writing, which is why Picard says that it's not a very promising start. San Jose State University holds an annual Bulwer-Lytton Fiction Contest, in which entrants are encouraged to come up with equally humorously incompetent opening sentences.
The mission patch on Col. Ritchie's (the dead astronaut) uniform, found in the wardrobe, is that of Apollo 17 with the word 'Apollo' obscured.
A scene in the casino contains a homage to The Questor Tapes (1974). Questor was one of several of Gene Roddenberry's failed attempts to launch a new series in the 1970s, and Data, for all intents and purposes, is a newer version of Questor. In the movie, while playing craps at a casino, Questor realizes the dice are slightly misshaped and physically adjusts them with his hands so that they will roll correctly, just as Data does in this.
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