system
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human
If there are an infinite number of prime numbers, why can it not be proven that there are an infinite number of twin prime numbers?

I was just watching a video about primes and twin primes, and it was said that there is no proof that there are an infinite number of twin primes. Why is that the case? If there are an infinite number of prime numbers, then shouldn't each prime number have a twin being the next prime number above it, with the chain going on infinitely? Sort of like a weird inductive proof?
gpt
Twin primes are not a prime and the next prime above it.

Twin primes are prime numbers that differ by 2.  So, for example, 11 and 13 are twin primes, 17 and 19 are twin primes, but 37 and 41 are not.

We do know that primes get, on average, sparser as one gets to larger and larger numbers.  As you say, whether there are still an infinite number of cases where the next prime is just 2 greater than the previous prime remains an open question, but recent progress has shown that there are infinitely many primes separated by 246 or less.