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

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?

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 $$3$$.
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).