I did this on paper and finally remembered to enter my answers to confirm they were correct. This actually took me around two hours, including the time necessary to look up "semiprime" and probably some other math words
Prime whose first three digits are p^q and whose last three digits are q^p (for primes p, q whose value is left as an exercise for the reader)
8A
69 cubed (nice)
9A
Largest prime number to appear on the Billboard Hot 100 charts [citation needed]
11A
Hex code for the shade of green whose RGB code is (65,132,2)
12A
___ and its Mysterious Recurring Decimals (Numberphile video about the largest perfect power that fits in the space for this answer)
13A
My age, in days, on my 20th birthday
Down
1D
THX ___ (George Lucas film referenced in many other George Lucas works)
2D
907 squared
3D
Multiple of XI (except not in Roman numerals; also, hope you learned the trick for checking divisibility by eleven, which is similar to the trick for nine except with an alternating sum, because eleven is one more than ten while nine is one less than ten)
4D
Phone number that, on "Scrubs", could (aptly) let you call Turk (but without the extra number at the end, though J.D. says "I'll always dial the 'K' for you")
6D
When halved (?!?), smallest semiprime whose prime factors are non-trivial reversals of each other
7D
Number that equals p^8 - p^4 for some prime p (although that function only has a six-digit output for one (1) input, so I didn't need to specify primality here, I guess)
10D
Number that can be factored as p(p-2), where p is a two-digit prime