Can anyone clarify why induction method fails for this conjecture?


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    $\begingroup$ One should think it fails because Goldbach's Conjecture is unsolved. $\endgroup$ – rschwieb Jun 7 '12 at 11:37
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    $\begingroup$ Have you tried to apply induction? If you try you should immediately detect the difficulties. $\endgroup$ – rschwieb Jun 7 '12 at 11:37
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    $\begingroup$ You can try it. If you fail you answered your own question. If you don't fail you will be pretty famous soon. Sounds like a win-win. $\endgroup$ – Simon Markett Jun 7 '12 at 11:40
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    $\begingroup$ ok. Will try to apply. :) $\endgroup$ – john Jun 7 '12 at 11:42
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    $\begingroup$ Boring Comment: if $p$ is prime, then $p+2$ is not prime in general (even if $p$ is odd). You should be able to see this is the case if you list the first few prime numbers. (Hint: "Few" $\geq 4$ if you restrict to odd prime numbers.) If $p$ and $p+2$ are prime numbers, then we say that the pair $(p,p+2)$ is a twin prime. An open problem is to determine whether or not there are infinitely many twin primes. $\endgroup$ – Amitesh Datta Jun 7 '12 at 12:06

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.

  • $\begingroup$ Why can't we say that we know that n+2 is even, so the induction hypothesis applies, so the conjecture being true for n (n is even) implies that it's true for n+2 since n+2 is also even? $\endgroup$ – Seth Taddiken Nov 5 '18 at 5:23
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    $\begingroup$ The induction hypothesis in my proof is that the conjecture is true for every even number up to $n$. You cannot immediately conclude from this that it is true for $n+2$ also. $\endgroup$ – MJD Nov 7 '18 at 15:08

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