Why proof by induction fails for Goldbach's conjecture?

Question

Can anyone clarify why induction method fails for this conjecture?

Answer 1

Let's prove the conjecture by induction.

Claim: For every even number $n≥4$, there exist primes $p$ and $q$ such that $p+q=n$.

Base case: $n=4$. Let $p=q=2$.

Induction step: Say that we know that the claim is true for every even number $k ≤ n$. We would like to prove that it is true for $n+2$ as well.

We have available for each even number $k≤n$ two primes, $p(k)$ and $q(k)$, with $p(k)+q(k) = k$. We need to find prime numbers $p$ and $q$ with $p+q = n+2$.

At this point I do not know how to proceed. Please help me out. How can I construct the desired $p$ and $q$ here?

Perhaps it can be done. But as far as I know nobody has yet thought of a way to do it.