Properties of prime numbers proven as finite Many modern proofs/conjectures concerning prime numbers (twin primes, infinite primes not containing one specific digit) intend to show that some property of prime numbers is infinite.
Are there any similar, nontrivial properties of prime numbers that have been proven to only appear within the first n, but not the rest (finite)? As a rough guideline to nontriviality, there should be at least ~3 somewhat spread out instances of the property that discontinue after reaching some number above ~50. The property cannot solely involve a simple operator such as < or >, and ideally should not involve repeated deletion of digits (truncatable primes) as this makes proof via exhaustion easier.
 A: While this assertion is still a conjecture rather than a proven statement, it's otherwise a classic example of exactly what the OP is asking:
We believe that there are only finitely many Fermat primes, that is, primes that are one more than a power of 2. Indeed, we believe that the only Fermat primes are 3, 5, 17, 257, and 65537.
(As a consequence, there would be only finitely many odd numbers $n$ such that a regular $n$-gon can be constructed with a ruler and compass.)
A: Some possible solutions after digging into this problem a bit deeper:

*

*Primes where you can repeatedly delete any number and still have a prime:
2, 3, 5, 7, 23, 37, 53 and 73


*Left truncatable primes


*Right truncatable primes


*Left and right truncatable primes
https://en.wikipedia.org/wiki/Truncatable_prime
However, all of these involve some sort of deletion of numbers, which make proof via exhaustion easy. The most interesting solution would not involve deletion.
EDIT with sufficient answer:
Upon consulting Wikipedia's list of prime numbers, it appears a few more candidate sequences exist including supersingular primes, minimal primes, Wilson primes, Wolstenholme primes, Fermat primes, and Stern primes.
A: There are some easy examples, as:
The prime $2$ is the only even prime.
The only primes that are consecutive integers are $2$ and $3$.
The only triple of primes differing by $2$ (that is: $p$, $p+2$ and $p+4$ are primes) is $(3,5,7)$.
The only prime of the form $n^4+4$ is $5$.
These are very elementary examples. I think the question is interesting. It would be nice to have a less trivial example of a theorem that is valid only for a finite number of primes.
