Ancient Greeks conjectured that there are infinitely many pairs of primes which differ by 2 (twin primes). A natural widely believed generalization is that 2 may be replaced by every even number. Moreover, it is expected that for every integer $T>0$ there are about $C n/\log^2 n$ numbers $p\leqslant n$ for which $p$ and $p+2T$ are both primes.

What is proved is that there are infinitely many pairs of primes with the same difference (Zhang and beyond). But it is not known for difference 2 or 4.

Ancient Greeks conjectured that there are infinitely many pairs of primes which differ by 2 (twin primes). A natural widely believed generalization is that 2 may be replaced by every even number. Moreover, it is expected that for every integer $T>0$ there are about $C n/\log^2 n$ numbers $p\leqslant n$ for which $p$ and $p+2T$ are both primes.

What is proved is that there are infinitely many pairs of primes with the same difference (Zhang and beyond). But it is not known for difference 2 or 4. If you (or chatgpt) need only one pair, this is certainly checked for all not too large integers, but I am afraid that still open for large enough integers.