A relation of prime numbers : $p_m + p_n ± 1, m,n ≠ 2$ is a prime where $p_r$ is a prime.

Question

Using a table of primes I noted the following pattern: $$p_2 + p_3 - 1 = p_4$$ $$p_3 + p_4 - 1 = p_5$$ $$p_3 + p_4 + 1 = p_6$$ $$p_4 + p_5 - 1 = p_7$$ $$.……………$$ But the pattern failed when, $$p_8 + p_9 + 1 = p_{14}$$ $$p_9 + p_{10} + 1 = p_{16}$$ However I could recover $p_{15}$ as, $$p_{13} + p_4 - 1 = p_{15}$$ Gradually I came to recognize a possible theorem: At least one of $p_m + p_n ± 1, m,n ≠ 2$ is necessarily a prime where $p_r$ is a prime. Can this be proved in any way?

Answer 1

Note that $26=7+19$ and neither of $26\pm 1$ is prime.

Indeed, if Goldbach's conjecture is right, there will be infinitely many counterexamples, because the even numbers $30n\pm 4$, for example, will be the sum of two odd primes, while $30n\pm 5$ is divisible by $5$ and $30n\pm 3$ is divisible by $3$.

There are many counterexamples anyway, because Goldbach's conjecture has been numerically verified for millions of values.

Answer 2

This is false.

Take $n=2$ or $p_n = 3$ your theorem says that for every prime either $p+2$ or $p+4$ is a prime which is definitely false. Take $p=23$ for example (I think this is the smallest counterexample).