The Weak and Strong Goldbach Conjectures

imagesIn 1742 the German mathematician Christian Goldbach wrote a letter to Leonhard Euler proposing two problems that, until recently, have seen little progress.  In modern terms, the problems are identified as either the Weak or Strong Goldbach Conjecture.  The Strong Goldbach Conjecture is the statement that every even integer greater than 2 is the sum of two prime numbers.  Similarly, the modern version of the Weak conjecture can be stated as every odd integer greater than 5 is the sum of three odd primes.

In his groundbreaking work earlier this year, the Peruvian mathematician Harald Helfgott (with much acceptance in the math community) announced that he had proved the Weak conjecture.  Fascinated by his result, I sent him an email of congratulations with a related question involving the Strong conjecture.  The following was his reply:

Dear Avery,

Thank you for your email. The fact that “three implies four” (so to speak)
was already known (and, as you can see, very easy to prove). I believe the
strong conjecture is much, much, much harder.

All the best
Harald Helfgott

 I too agree that the Strong conjecture is, “much,much,much harder”.      On May 13, 2013, the renowned mathematician Terrence Tao of UCLA released the following statement on the internet in regards to Helfgott’s result:

Busy day in analytic number theory; Harald Helfgott has complemented his previous paper (obtaining minor arc estimates for the odd Goldbach problem) with major arc estimates, thus finally obtaining an unconditional proof of the odd Goldbach conjecture that every odd number greater than five is the sum of three primes….. As with virtually all successful partial results on the Goldbach problem, the argument proceeds by the Hardy-Littlewood-Vinogradov circle method; the challenge is to make all the estimates completely effective and to optimise all parameters (which, among other things, requires a certain amount of computer-assisted computation). [EDIT: the proof also relies on extensive numerical verifications of GRH that were performed by David Platt.]”

Unfortunately, Tao further stated that it would be unlikely that the Hardy-Littlewood Vinogradov Circle Method and Helfgott’s result could be used to prove the Strong conjecture.  Therefore the mystery of the Strong conjecture has currently escaped all attempts of proof.    Computationally, it has been shown true for every even integer greater than 2 into the trillions.   It would be clever indeed  for some mathematician to show that the Strong conjecture follows trivially from the Weak.  Do you think you can do it?

About Avery Carr

I am a grad student studying mathematics at Emporia State University. I obtained a B.S. in Mathematics from the University of Memphis in 2010 where I served as President of the math club my senior year. My research interests include Graph Theory, Number Theory, Recreational Mathematics, Mathematical Physics, and Operator Theory.
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6 Responses to The Weak and Strong Goldbach Conjectures

  1. Shijie Gu says:

    Thank you for sharing this important consequence in number theory. really enjoy it. I have a quick question, Avery. Comparing with the work of Chen’s 1+2 (1 prime + semiprime(product of two primes)), whose works are closer to the strong conjecture? Helfgott or Chen?

  2. Avery Carr says:


    Thank you for your comment and question. That is an interesting question. Unless Dr. Helfgott is working with other tools for the Strong Goldbach Conjecture, Dr. Tao stated that Dr. Helfgott’s current result is unlikely to be used to prove the Strong Conjecture. However, unlikely and improbable does not mean impossible. But I do give a lot of credibility to Dr. Tao’s mathematical intuition, given his own epic contributions to mathematics. I think Chen’s Theorem is a great result. If I were going with my own intuition limited to this conversation, I would say the Dr. Chen Jingrun may be closer, based on comparing both theorems and taking Dr. Tao’s comment into consideration. But then again, you never know with these things. Dr. Helfgott could produce a proof of the Strong Conjecture tomorrow using some other tool. Thank you again for your thought provoking question. It was a needed addition to this article.

  3. Frank Vega says:

    This is one of the most difficult problems in mathematics. Harald Helfgott and Terence Tao have made a great contribution. Even though, it’s much harder the strong conjecture, I think the humanity is closer in the solution of this oldest conjecture. I think one important step in this way would be to prove the ternary conjecture (which was the original conjecture of Goldbach):”… at least it seems that every number that is greater than 2 is the sum of three primes” (considering 1 as a possible candidate, although it is not considered anymore as a prime). It seems to me this could be a good start to face the final solution definitely.

  4. Frank Vega says:

    Carr, I have submiited a possible solution to the ternary conjecture to Annals on May. I have replaced it and they have accepted my revisions until now. If you wish to share your opinion with me, I would thank you. I will share you my preprint where is the version 4, which is the actual version that I sent to Annals this month after I fixed all the trivial details:


  5. Frank Vega says:

    The proof of Harald Helfgott implies the ternary conjecture. I was wrong in my comments. Indeed, I was trying to prove that any even number greater than two could be written as the sum of one prime with another or the number 1. However, the proof that I sent to Annals has some flaws. Nevertheless, I extracted the best arguments of this paper and tried to do another paper with more serious arguments. I decided to send it again to Annals today. The paper that I sent is not a definitive solution, but it could help to find the final solution on this unsolved problem. Furthermore, it could help to see beyond the wall on this oldest mystery if it might be published in that journal. I hope so…

  6. janet middlemiss says:

    No one states for the weak problem (or strong) if the primes can be repeated, or must each – say weak – be the sum of three primes which are all different from one another?

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