A 17-word Facebook reminder contributed to substantial increases in online voter registration across the country, according to top election officials. At least nine secretaries of state have credited the social network’s voter registration reminder, displayed for four days in September, with boosting sign-ups, in some cases by considerable amounts. Data from nine other states show that registrations rose drastically on the first day of the campaign compared with the day before. “Facebook clearly moved the needle in a significant way,” Alex Padilla, California’s secretary of state, said in an interview on Tuesday.