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Old 2014-12-01, 15:19   #1
cuBerBruce
 
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Aug 2012
Mass., USA

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Default Twin Prime Days, Prime Day Clusters

Happy Twin Prime Days, 20141201 and 20141203 !

1. How many twin prime day pairs are there in all this calendar year (2014)?
2. When will the next pair of twin prime days occur?
3. When will be the next occurrence of two consecutive days that are both prime days?
4. When will be the next time three prime days occur within a 4-day stretch?
5. When will be the next time 5 prime days occur within a 10-day stretch?
6. When will be the next time 5 prime days occur within a 10-day stretch, with 4 of them being in the same month?

Two additional questions, not related to clusters of prime days:
7. On what day of the year will the next Mersenne prime day occur? (Assume that the Gregorian calendar will continue on forever. Never mind the fact that the sun will no longer be shining.)
8. What will be the next prime day that consists of only the same digit repeated some number of times?

Assume the Gregorian calendar and dates are interpreted as number using yyyymmdd format. The year can be more than 4 digits, if needed. Numbers (including yyyy, mm, and dd components) are to be interpreted as decimal only. The mm and dd components must use exactly two digits, and so must include a leading 0 if less than 10. Only valid combinations of mm and dd can be used, and February 29 (0229) must only be used with actual leap years.
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