Twin Prime Days, Prime Day Clusters

[cuBerBruce]

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.

[TimSorbet]

1. Four: 20140301 and 20140303, 20140829 and 20140831, 20140907 and 20140909, 20141201 and 20141203
2. 20160401 and 20160403
3. 20170831 and 20170901
4. 20170831, 20170901, 20170903
5. 20800529, 20800531, 20800601, 20800603, 20800607
6. 30830131, 30830201, 30830203, 30830207, 30830209
7. October 7th. This comes from M2203, whose full 2^p-1 value taken as ...yyyyymmdd is a 660 digit year with 1007 as mmdd. Some fun trivia: this prime was discovered on its "date" (just the wrong year), October 7th, 1952. The other Mersenne prime days are from M4423 (0607) and M37156667 (0927). Neither of them share the property of being found on their "day". And if you try to take the p as the date instead of 2^p-1, no Mersenne prime days are known. We'll probably discover one some day.
8. 1111111111111111111 (111 trillion years from now)

[cuBerBruce]

Congratulations to Mini-Geek for getting 7 out of 8 correct!

Only incorrect answer was for #7. Perhaps Mini-Geek was thinking of Mersenne prime exponents, not Mersenne primes? (Yes, I see that his answer indicates he was.)

OK, Mini-Geek has now changed his answer to #7, and all eight answers are now correct!

[TimSorbet]

Originally Posted by cuBerBruce

OK, Mini-Geek has now changed his answer to #7, and all eight answers are now correct!

Originally Posted by Mini-Geek

And if you try to take the p as the date instead of 2^p-1, no Mersenne prime days are known. We'll probably discover one some day.

No [I]future[/I] Mersenne prime exponent days, I should say. 11213 and 110503 are the only known ones, both from the first century AD (and assuming the leading zeroes on the year aren't important).