According to the Gregorian calendar, 97 leap days must be inserted over 400 consecutive years. After how many years will there be a 1-day difference between the Gregorian calendar and the "precise" calendar, if the length of a year is 365 days 5 hours 48 minutes 46 seconds?
Regardless of the number of years, the Gregorian calendar offers 97 days every 400 years, which is $97 \cdot 24=2328$ hours, compared to 5 hours, 48 minutes, and 46 seconds 400 times over. It is reasonable to count the minutes as $\frac{4}{5}$ of an hour, thus in 400 years, $400 \cdot\left(5+\frac{4}{5}\right)=$ 2320 hours are obtained, which is 8 hours less than 97 days. For the benefit of the Gregorian calendar, therefore, 8 hours $=28800 \mathrm{~s}$ stands against $400 \cdot 46=17400$ seconds. Thus, the excess every 400 years is $28800-17400=10400$ seconds. To make a full day out of this, that is, 86400 seconds, $\frac{400 \cdot 86400}{10400} \approx 3323$ years are needed.