^I just went over the transcript of the movie, and the number 15 shows up several times. The second line in the movie is "Leaving Sector Fourteen for Sector Fifteen," in the simulator scene with Kirk watching. Then, just after Spock gives Kirk the book, a PA voice pages Spock to say his shuttle leaves in 15 minutes. There's a "Fifteen seconds" uttered by a computer countdown during the prefix code scene. And before Kirk beams down to Regula I, Saavik advises him that General Order 15 requires an armed escort. So maybe he just had the number 15 on the brain and misspoke?
(Certainly the screenwriters seems to have the number 15 on the brain.)