For some uncanny reason, when I came up with my new totally failed upper bound for the number of repetitions, my brain seemed to imagine that the primes in the 3rd and 4th quarters were spaced out the same way as they are in the first quarter. I have spent a day and a half trying to fix the problem but to no avail as yet. Many thanks for your computer simulations which got me thinking about where I have made errors. I will spend a few more days rummaging about how to fix this problem which might mean thinking about a slightly modified approach. At the back of my mind, I know the truth of the conjecture is out there.

[QUOTE=Awojobi;521174]For some uncanny reason, when I came up with my new totally failed upper bound for the number of repetitions, my brain seemed to imagine that the primes in the 3rd and 4th quarters were spaced out the same way as they are in the first quarter. I have spent a day and a half trying to fix the problem but to no avail as yet. Many thanks for your computer simulations which got me thinking about where I have made errors. I will spend a few more days rummaging about how to fix this problem which might mean thinking about a slightly modified approach. At the back of my mind, I know the truth of the conjecture is out there.[/QUOTE]Why don't you also think deeply about the content of my post and then write a post of your own which explains what, if anything, you may have learnt from it?

[QUOTE=xilman;521183]Why don't you also think deeply about the content of my post and then write a post of your own which explains what, if anything, you may have learnt from it?[/QUOTE]
Man, don't jinx it! Even the fact that he have seen, and admitted, which is totally different of seeing it only, (one of) the mistake(s), it is a HUGE step forward. The cause is not lost. He may turn out being some great mathematician over the years... :smile:

[QUOTE=LaurV;521404]Man, don't jinx it! Even the fact that he have seen, and admitted, which is totally different of seeing it only, (one of) the mistake(s), it is a HUGE step forward. The cause is not lost. He may turn out being some great mathematician over the years... :smile:[/QUOTE]

Yes, and I do not wish to give up on him/her. Awojobi might be a beginner, hence he/she might not understand what constitutes a proof. We just need to be more patient with him/her. He/she might seem to not address some of the feedback given but he/she seeks to clarify doubts when needed. At the very least, he/she is willing to learn. Compare this to samuel, who was arrogant and stuck to his own views even when 100% concrete evidence showed otherwise.

Admittedly, I was under pressure from serious mathematicians to stop arguing with Awojobi, hence there was a point where I lost patience of him/her.

[QUOTE=2M215856352p1;521510]Yes, and I do not wish to give up on him/her. Awojobi might be a beginner, hence he/she might not understand what constitutes a proof. We just need to be more patient with him/her. He/she might seem to not address some of the feedback given but he/she seeks to clarify doubts when needed. At the very least, he/she is willing to learn. Admittedly, I was under pressure from serious mathematicians to stop arguing with Awojobi, hence there was a point where I lost patience of him/her.[/QUOTE]
Which was why I posted a [URL="https://mersenneforum.org/showpost.php?p=520827&postcount=143"]relevant document[/URL] sent to me by a serious mathematician.

PROOF OF GOLDBACH’S CONJECTURE

Goldbach’s conjecture states that every even integer > 2 is the sum of 2 primes. An example is 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53. An alternative statement of the conjecture is as follows. Every integer > 3 is the arithmetic mean of 2 primes. This means that every integer > 3 is midway between 2 primes. For instance, 9 is midway between 5 and 13. 9 is also midway between 7 and 11.

Imagine a finite part of the number line up to some integer, N. Let this number line be divided up into roughly 4 equal parts. If it can be shown that every integer in the 2nd quarter of this number line is the arithmetic mean of 2 primes, then Goldbach’s conjecture is proved. So prime number 3 (in the 1st quarter of the number line) and primes mainly in the 3rd and 4th quarters of the number line will produce arithmetic means in the 2nd quarter of the number line. The next prime number, 5 (in the 1st quarter of the number line) and primes mainly in the 3rd and 4th quarters of the number line will produce arithmetic means in the 2nd quarter of the number line. Some of these arithmetic means will be repetitions of those produced using prime number 3. The number of new arithmetic means produced by prime number 5 in the 2nd quarter of the number line is equal to the number of arithmetic means produced by prime number 5 minus the number of twin primes (5 – 3 = 2) mainly in the 3rd and 4th quarters of the number line. The next prime number, 7 (in the 1st quarter of the number line) and primes mainly in the 3rd and 4th quarters of the number line will produce arithmetic means in the 2nd quarter of the number line. Some of these arithmetic means will be repetitions of those produced using prime number 3 and prime number 5. The number of new arithmetic means produced by prime number 7 in the 2nd quarter of the number line is equal to the number of arithmetic means produced by prime number 7 minus the number of twin primes (5 – 3 = 2), mainly in the 3rd and 4th quarters of the number line, minus the number of primes that differ by 4 (7 – 3 = 4), mainly in the 3rd and 4th quarters of the number line. This process carries on until all arithmetic means in the 2nd quarter are produced and any further process will produce only repetitions of arithmetic means.
The phrase, ‘mainly in the 3rd and 4th quarters of the number line’, is used because primes in the second quarter will also be involved in producing arithmetic means in the second quarter.

The number of primes less than N is approximately equal to N / log e N. This is a lower bound for N ≥ 17. Because log e 2 is small compared to log e N, the number of primes in the second half of the number line is approximately equal to (N/2)/log e (N/2). Using the same argument, the number of primes in any part of the number line with N/2 consecutive integers contained in the 2nd, 3rd and 4th quarters is approximately equal to
(N/2)/log e (N/2). Considering the discussion in the previous section, each prime will produce approximately (N/2)/log e (N/2) arithmetic means. If there were no repetitions, it would take approximately [log e (N/2)]/2 primes in the first quarter to produce all N/4 integers in the 2nd quarter of the number line. This is a relatively small number of primes when compared with the number of primes in the 1st quarter of the number line. This begins to give the idea that only a small fraction of the number of primes in the 1st quarter of the number line will be required to achieve Goldbach's conjecture. Apart from prime number 3, the other primes used in the 1st quarter of the number line will produce repetitions of some of the arithmetic means produced by primes used prior to the current prime. Because the first quarter has an abundance of approximately (N/4)/log e (N/4) primes, every integer in the 2nd quarter would be produced at some point in the process. This statement will be justified. Let's use prime number 19 as an example as it produces approximately (N/2)/log e (N/2) arithmetic means in the second quarter, of which some will be repetitions of arithmetic means produced by prime numbers 3, 5, 7, 11, 13 and 17. A repetition will occur when a prime in the 2nd, 3rd or 4th quarter has a prime difference of 19-17=2, 19-13=6, 19-11=8, 19-7=12, 19-5=14 or 19-3=16 with one or more primes in the 2nd, 3rd or 4th quarter. The maximum number of prime differences, 2, 4, 6, 8, 10, 12 etc. that can occur in the process is half the number of integers in the 1st quarter i.e. N/8.This is a large number of prime differences and so there is ample opportunity for all the N/4 integers in the second quarter to be produced by the process described since a small fraction of the total number of N/8 prime differences is considered for each prime in the 1st quarter that produces arithmetic means. So only a small fraction of the approximately (N/4)/log e (N/4) primes in the first quarter is required to produce all the N/4 integers in the 2nd quarter. Goldbach's conjecture is proved.

[QUOTE=Awojobi;525983]... approximately ... approximately ... approximately ... approximately ... approximately ... approximately ... approximately ... approximately ...[/QUOTE]Really? Everything is approximate, nothing definite. Where is the [i]proof[/i]?

Everything is correct, except the phrase

[B]"This process carries on until all arithmetic means in the 2nd quarter are produced and any further process will produce only repetitions of arithmetic means."[/B]

This phrase, as many people here already argued, has to be reformulated:

[B]"This process carries on until [COLOR=SeaGreen](A)[/COLOR] all arithmetic means in the 2nd quarter are produced and any further process will produce only repetitions of arithmetic means,[COLOR=Red] [COLOR=seagreen]or (B) [/COLOR]the first prime is >N/2, whichever comes first.[/COLOR]"[/B]

If you can prove that the red text never happens, then Goldbach is proved.

Until now, we didn't see any proof, but we have seen a lot of accusations of the fact that we want to "steal" your proof, which you don't yet have. We are highly tempted to close all your threads related to Goldbach until you come out with something more palpable or you give up, but we are afraid that other moderators here will kick us in the nose... :leaving:

I spent weeks trying to come up with a satisfactory proof. This proof explains the mechanics of what is really going on concerning how it is inevitable that all arithmetic means will be produced. No need going over the re- branded proof again. You read it again without bias. Look at the first set of computer simulations a helpful poster produced weeks ago and you will see that all arithmetic means are produced relatively quickly, i.e. a very small fraction of primes in the first quarter are needed to achieve Goldbach's conjecture.

A pattern that holds true in the short term is no guarantee that it will hold in the long term. And infinity is a really looooong term.

Your proof is not satisfactory.

Let N be 10^123456+7891011, and let n=30*10^123454+3210, which is somewhere at the beginning of the second quarter, please show me the primes that cover n.

And then, tell us why do you need N and all the artifact of dividing into 4 intervals, etc., what plus of information or help does this bring? Couldn't you just pick a number, any number, and just show how this is covered by "symmetric" primes?

You don't even understand that what you do, is this: you take some range, split it in 4, take a number from the second quarter. To prove that this number is covered by (at least) one pair of symmetric primes (related to it), you need to use the fact that the Goldbach conjecture is true :razz: You make an affirmation which is equivalent with the conjecture itself, and decide that the conjecture is true because the affirmation you made is true. Nice...