According to the following, it doesn't hold for higher powers: https://oeis.org/A117757 After 1336, the numbers are: 4642, 16458, 59025 (NOT EVEN), 213922, 781924, 2879938 ....... a few more odd ones later on as well ....
There's a challenge - someone prove Trev's Theorem - a la Fermat's Last Theorem which remained unproven for years - and I couldn't follow the eventual proof anyway ;-)
Fascinating. I continued the pattern for a bit
1024-4096: 392 primes
4096-16384: 1336 primes
For the sake of accuracy if this is for work you are submitting, 4 to the power 0 is of course 1 not 0, but that doesn't change the results.
Sadly in my sleep I found myself working out differences between squares, and considering how primes tending to come in pairs, and formulated an idea- but I woke up with a headache and couldn't recall all the details.
There is also the Goldbach conjecture which seems simple on the face of it but hasn't been proven as far as I know. This says that every even number above 2 is the sum of two prime numbers
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