Twin Primes of the form $3k-1, 3k+1$

Question

I wanted to discuss something. Yesterday I thought about the twin prime conjecture and I constructed numbers of the form $$ 3k-1, 3k, 3k+1 $$ Then I proved with the help of quadratic reciprocity, that there are inifinitely many prime numbers of the form $3k+1$. I plugged in some even numbers for $k$ and I thought, that if $3k+1$ is prime, then $3k-1$ has to be prime, hasn't it? - No, because $3\cdot 12+1=37,$ which is prime but $35=5\cdot 7$ surely isn't. I was disappointed but nevertheless I figured that there are quite a lot of prime pairs $3k-1,3k+1$. Is there a way to prove or contradict this? If so, we would have infinitely many twin primes, wouldn't we? Maybe we have to find more conditions so that this pair of numbers is really prime.