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Why doesn't proof of the weak Goldbach conjecture also prove the strong Goldbach conjecture?

Actually I am referring to this link. My question is why the logic used in this question cannot be used to constitute a formal proof of the strong Goldbach conjecture? The comments made on that question were not very clear for understanding so I am asking this again and expecting a detailed and better explanation. I am willing to refine this question even more, if required.

EDIT

Thanks a lot @lulu for pointing that out. I concur that it's very vague and not a formal proof according to mathematical standards. But I would counter that as follows:

Here "O" represents any odd number greater than 5 and we are considering "a" to be an odd prime number (such "a" always exists in pair of three primes whose sum is an odd number).

Lowest possible value of "a" is 3 because below that we have the only even prime number i.e. 2 Then no matter what value the odd number "O" takes greater than 5 if we subtract 3 from it we get an even number.

Now the next odd number after 5 is 7 and subtracting 3 we get 7 - 3 = 4 (This is the smallest supporting example of the proof).

Then no matter what values "O" and "a" take as long as "O" is odd greater than 5 and "a" is odd prime greater than 2 their subtraction i.e. "O - a" will always result in an even number greater than or equal to 4.

And this is exactly what the strong Goldbach conjecture states - every even natural number "greater than 2" is the sum of two prime numbers.

Ok-Virus2237
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    Apparently not. I do not know why the strong Goldbach conjecture is so much more difficult. An analogue situation : We know that infinite many prime gaps do not exceed $246$ , but we cannot use this to prove the twin prime conjecture. – Peter Oct 08 '23 at 13:21
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    Well, the "proof" provided in that post is very vague. You would have to show that every even number arises as $O-a$ through that process, and that point is not addressed. – lulu Oct 08 '23 at 13:24
  • related – Peter Oct 08 '23 at 13:26
  • Well, the usage of the analogue situation is appreciated, and it does feel like "apparently not" because many mathematicians who worked on this conjecture concluded that it may require some new and novel method/technique to prove it instead of such fundamental properties. But originally my question was "why not?" and what is the issue in this logic? – Ok-Virus2237 Oct 08 '23 at 13:26
  • If you want to know where the flaw in a "proof" of the strong Goldbach conjecture is , summarize the argument here in the body of the question and do not make the question "link-dependent". Some users argue that even this would not be enough and demand exactly the step where you have doubts , but some do not agree this and will point out the flaw after a good summarize is shown. – Peter Oct 08 '23 at 13:30
  • @Peter Can you please help me out to not make the question "link-dependent" and to improve its readability? I would be grateful if you suggest me an edit from your side as I am a beginner on this forum. – Ok-Virus2237 Oct 08 '23 at 13:52
  • @lulu I have responded to you too by editing the original question and I await your further opinions. – Ok-Virus2237 Oct 08 '23 at 13:54
  • I don't see any improvement over the linked post. Nobody doubts that $O-a$ is an even number which is the sum of two primes. But so what? The goal of Goldbach is to show that every even number is the sum of two primes (or, in the case of $2$, prime itself), so to complete this argument you would need to prove that every even number arises this way. – lulu Oct 08 '23 at 14:46
  • To stress: it is easy to prove that infinitely many even integers are the sum of two primes. Indeed, for any prime $p$, $2p$ is the sum of two primes, as is $p+3$. – lulu Oct 08 '23 at 14:47
  • @lulu but is not it that every even number greater than 2 can be written in the form of O − a (where O is some odd > 5 and a is some prime > 2)? Like I may be very wrong so kindly explain. – Ok-Virus2237 Oct 08 '23 at 17:40
  • You have to prove that! That's the whole point. Neither the linked post, nor your version of it, even addresses the point, and it's the only point that matters. – lulu Oct 08 '23 at 23:29

2 Answers2

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Consider this list of numbers (known as Lourrran's numbers) : $O, 3, 5, 7, 11, 15, 23, 25,$ ... and all odd numbers greater than $25$.

Weak conjecture : Every odd number $n$ greater than $3$ can be written as a sum of 3 Lourrran's numbers.

Easy to check that this conjecture is correct.

Strong conjecture : Every even number greater than $6$ can be written as a sum of 2 Lourrran's numbers

Easy to check that this conjecture is false, there is no couple $(a,b)$ of Lourrran's numbers such that $a+b=24$.

Edit :

Of course, I have created those L-numbers just for this message.

You say :

when we have a suite of odd numbers (Primes numbers without 2 in your case),

knowing that any odd number can be written as a sum of 3 numbers from this suite

should imply that :

any even number can be written as a sum of 2 numbers from this suite.

I show that :

Based on a suite of odd numbers (see L-numbers),

This suite matches the first predicate (any odd number can be written as a sum of 3 L-numbers)

and this suite does not match the 2nd predicate.

So what you say is wrong.

Lourrran
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  • What is the definition of so-called "Lourrran's numbers" ? – Ok-Virus2237 Oct 08 '23 at 13:50
  • @Ok-Virus2237 Lourrran defines them right here. They seem to have been invented precisely to provide this answer to the question. – Ethan Bolker Oct 08 '23 at 13:54
  • @EthanBolker But how "Lourrran's numbers" are connected to the conjecture and how they answer the original question? I did not get that – Ok-Virus2237 Oct 08 '23 at 14:13
  • @Ok-Virus2237 It does not answer the original question. All it does is offer an alternative rather boring example in which the "weak conjecture" is true and the "strong" one is false. – Ethan Bolker Oct 08 '23 at 14:57
  • You might want to point out that the linked argument works equally well for your numbers. That is, write an odd number $O$ as the sum of three $L-$numbers, $O=a+b+c$. Unless $O=0$, at least one of $a,b,c$ is odd, so say that $a$ is odd. Then $O-a$ is an even number which is the sum of two $L-$numbers. But, this does not prove the strong conjecture (which indeed is not even true). – lulu Oct 08 '23 at 15:20
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    The answer demonstrates the flaw of the argument by showing an example where applying exactly this argument leads to a wrong conclusion. So how does this not answer the question ? (+1) – Peter Oct 08 '23 at 15:25
  • Well, although I want to appreciate that the analogy of L-numbers is nicely crafted out however the weak conjecture also has counterexamples. It states that every odd number n greater than 3 can be written as a sum of 3 L-numbers. And the next odd number after 3 is 5 which is a counterexample itself because I can't find any L-numbers which add up to 5 because the L-numbers below 5 are O and 3. – Ok-Virus2237 Oct 08 '23 at 17:54
  • I have edited my answer. – Lourrran Oct 08 '23 at 18:59
  • @Peter I disagree that this answer demonstrates the flaw of the argument. It certainly proves the argument wrong, but again I disagree that it demonstrates the specific flaw. I answered a bit more specifically in my answer, though I like this one too and is a perfectly correct answer to this question. – Snared Oct 08 '23 at 20:14
  • I believe it's more accurate to say that this answer demonstrates the existence of a flaw in the argument, but it doesn't demonstrate the flaw in the argument, which is an important distinction to make.. – Snared Oct 08 '23 at 21:54
  • @Snared : Agree, In such topics (wrong conjecture), it is often impossible to demonstrate where is the flaw. Often, we can say : what you say is not true neither false. So I did not search the specific moment where it falls from true to false. – Lourrran Oct 09 '23 at 00:01
  • @Lourrran I still claim that this example exactly shows where the proof breaks down. I prefer this answer since it actually shows that the argument can fail , and this shows that this possibility is there also in the case of the Goldbach conjecture. – Peter Oct 09 '23 at 08:17
  • how does this example show where the proof breaks down? It shows that the proof breaks down, yeah, but it doesn't show where the proof breaks down. – Snared Oct 09 '23 at 17:27
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Just because you know that every odd number $2n+1$ will write an even number $2k < 2n+1$ as a sum of two primes, that doesn't tell you that every even number $2u$ will be found as a $u=k$. The specific jump in logic that makes the proof wrong is the assumption of surjectivity of the conversion process. Indeed, every odd produces an even number written as a sum of two primes through this process, yes, but there was no proof that every even number will be discovered in this process.

Snared
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