Can anyone clarify why induction method fails for this conjecture?
Why proof by induction fails for Goldbach's conjecture?
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$\begingroup$
elementary-number-theory
prime-numbers
conjectures
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5Have you tried to apply induction? If you try you should immediately detect the difficulties. – 2012-06-07
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25You 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. – 2012-06-07
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1ok. Will try to apply. :) – 2012-06-07
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0If the conjecture is ever proved, it seems very likely that some sort of induction will be used somewhere. – 2012-06-07
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2***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. – 2012-06-07
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0What makes you think it fails? ${}{}{}$ – 2012-06-07
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2One should think it fails because Goldbach's Conjecture is unsolved. – 2012-06-07
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1Not so. It might work, but nobody has yet been clever enough to see how to carry out the induction step. – 2012-06-07
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1Leaked from the $\epsilon_0$ edition of *Proofs from the Book:* "I have discovered a truly marvelous proof of Goldbach's conjecture. But, alas, induction up to $\epsilon_0$ is too large to fit in Peano arithmetic." – 2012-06-07
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1I have that edition, it is superb. Much better than the edition preceding it! – 2012-06-07
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1It might also be possible to prove Goldbach's conjecture using modular representation theory of finite groups, but then again it might not. Why *should* one think a proof by induction is possible? – 2012-06-07
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1As far as I know, there is no proof that induction fails. – 2012-06-07
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0One does not need a proof that certain approaches fail to come to the conclusion that certain approaches may well be a waste of time. – 2012-06-07
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0@Number Do you want to say that a proof via transfinite induction exists ? Or do I misunderstand your comment ? Or do you refer to Fermat with his last theorem ? – 2018-08-09