A conjecture about the form of some prime numbers

Question

Let $k$ be an odd number of the form $k=2p+1$ ,where $p$ denote any prime number, then it is true that for each number $k$ at least one of $6k-1$, $6k+1$ gives a prime number.

Can someone prove or disprove this statement?

Answer 1

$p = 59 \implies k = 2p + 1 = 119$. Neither $6k+1 = 715$ nor $6k-1 = 713$ is prime. Some other counter examples are:

59 83 89 103 109 137 139 149 151 163 193 239 269 281

Answer 2

You got confused with your quantifiers, but if your conjecture is what I guess it is, then the first five counterexamples are $p=$ 59,83,89,103,109.