Direct Proof of Goldbach's Conjecture

Many attempts to prove than any even number (>3) is the sum of two primes try to find a contradiction, but I believe a direct proof is more likely to be the right approach. I have attached such an approach. It uses mod(6) to show for all 2n, (n>1) there are only three patterns of eligible pairs. And, these pairs can be examined to determine a prime-limit above which a prime pair must exist and below which a 2n could have no prime pairs.

No real surprise - but no 2n (n>1) can have a number of primes below its prime-limit. In fact, this proof shows that all 2n (n>6) must have TWO prime pairs.

Note: the "discussion" at the end is optional reading. It is not necessary for the proof, but it was useful to me in getting a count of composites for any (large 2n). I developed this approach, as a substitute for p-series arithmetic progressions, by identifying composites by their lowest common prime.

(See attached file.)

 

I am not intelligent enough to answer as to the validity of this work. It is clearly well done however. What my concerns are..

What is the point?-ergo the ramifications on the real world.

What is the philosophy?-ergo the ramifications on the absract world.

Surely there need not be any...as mathematics has had "proofs" before without philosophical and real world consequences. But if there are none, then this is only good for you, and your community. I personally believe that what we do should lift up all of us. That what we do should in some way tie back into the concept of love.

This said, I hope one of the "real" mathematicians on this site, take the time to read this and give you their thoughts. I know that it is most unlikely... as it takes time. One of the most precious things that we have. Perhaps you might offer an incentive for their time. You clearly deserve it.

Well done, have patience, and perseverance. Try other web sites, if this one fails you.

 

Two questions.

If you do not have a PhD in math, then if your proof is good, why are you not delivering it in a PhD thesis are a top tier university?

If you do have a PhD in math, then why are you not presenting it for consideration to your colleagues and to a top tier refereed mathematics journal?

Just asking.

 

Phrzby

A Phd is not required. Pythagoras had no phd. Descartes had no phd. This list goes on, and on. If you are not interested don't say anything at all. Do you have to re-watch the movie Bambi again, so that you might learn simple morality?

 

Your answer is immature and does not recognize how serious math is done.

My PhD comment was serious. If he has no PhD - this should earn it. That should be a big deal.

If he has one - then he should refer this to a refereed journal, rather than just a bunch of people on a website.

 

On the contrary. It pointed out, that for a FACT, a phd is not necessary. Did the Wright brothers know anything of aviation when they became the first to fly? There... another absract example for a mind lost in analytics. No one in this world holds your phd to value except others like you. And you are by far in the minority.

 

Geez. I never said it was necessary or anything like that.

My point is - if someone is doing PhD worthy work - why would he not want to earn his PhD?

If he has a PhD - then presumably this is high quality research that should be published in refereed journals.

 

Because one of the first things that happens to people in the academic world, is that they lose their sense of individual creativity. The academic world takes "diamonds" in the ruff and then it polishes them. You cannot polish a gem without cutting it. Of forcing it to lose some of what it already is. Additionally the academic world teaches "its" kind, ergo you. That people who have not gone through the system are incapable of possessing intellect.

 

Conway, stop being a toddler.

You wrote:
"I personally believe that what we do should lift up all of us. That what we do should in some way tie back into the concept of love."

This is a mathematics forum.
This is about mathematics, not about religion, philosophy, love or personal believes.

Do you also believe that mathematics should be usefull to the LGBTAAA++ community or else maths is transphobic? Please grow up.

 

I believe that philosophy and love is relative to all things. Not mathematics. You may call me names all you wish. I do not care.

You should know however..."logic", and "reason" are insignificant qualities when compared to the concept of humility, kindness, and above all love.

Be kind...you might like it.

 

Conway: this forum is not about "the concept of humility, kindness, and above all love."
This forum is about mathematics.
Your remark "Do you have to re-watch the movie Bambi again, so that you might learn simple morality?" has nothing to do with mathematics and is very rude.
If you have nothing usefull to say, then please don't say anything.
And please: start growing up.


Phrzby Phil: ignore the attention-seeking toddler.
You are right with your PhD-comment.

 

My PhD is in (Software) Engineering Management, and a version of this proof is currently being reviewed by the Mathematics Department at UTSA (my Alma Mater).

Here's an updated version generated from a LaTeX complier (a review and publishing app).

The "worth of this work for society" is probably minimal. Who cares if I and I is two? Why climb a mountain - just for the view? Why do a crossword? When you are through, what do you have? A smile in your heart! Seeing God's world fit together and having a place in that mathematical lattice is very comforting - and worth sharing with anyone who cares to share such things.

 

If you're into a Concluding/Eureka statement, this is as close to one as I can make on this topic:

Take any 2n circle and inscribe it with all of its prime factor regular polygons each with a node at position n. Eliminate their nodes and what remains (of the 2n set) are "horizontal" pairs of nodes with more primes than nonprimes. So, some primes must pair together.

 

Very well stated! I agree. Yet it remains if climbing a mountain, makes others, along with you better. Than it is better than making just you better! I suggest that if you look hard enough you would find that it does help the world, other than just mathematicians. But you must look with philosophical eyes, not mathematical eyes. Many times mathematics had no "point". At the time. Yet given long enough we found how it did indeed make others better. This is where I would point you to. I stand by to help you in this regard.

Your efforts to further our understanding of math is GREATLY appreciated!