What is "Goldbach weak conjencture"?

Goldbach's weak conjecture is simplified version of Goldbach's strong conjecture,which was formed in 1742 by Leonharda Eulera. This is the problem of Number theory. In 1742 Christian Goldbach in letter to Euler stated the hypothesis that:

After reading the letter Euler concluded that this hypotesis can be simlified to the following:

Examples for "Goldbach's strong conjecture":

Of course for a certain number ther could be more than one examples:

Even that the hypothesis was formed by Euler, we call it "Goldbach's conjencture" which is rare in the world of mathematicians.

Here You can find the reprint of the Goldbach's letter to Euler from 7th of June 1742, in which he forms the hypothesis. On math.dartmouth.edu site You can also find all the other letters of these famous mathematicians.

Going back to the weak conjencture, we should cite it literally:

Examples for "Goldbach's weak conjecture":

Of course similarly to the "Strong" version, there can be more than one possibility here also:

At the international mathematicians congress in 1900, David Hilbert announced his famous list of 23 mathematical riddles, which we now call "Hilbert's problems". Most likely, even the creator of the list didn't know the problematics of the issues and that even after hundred years of science development there will be questions that have no answers. The riddlenumber 8 is the Goldbach's Conjencture and Riemann Hypothesis - these are still open problems.

Even after 260 years there is no final conclusions of these problems.

If the "Goldbach's strong conjencture" will be found to be true, then the weak conjencture would also be true. Then you only need to add 3 to the even number greater than 5 and present received odd number in accordance with the "Goldbach's strong conjencture".

Example:

"Goldbach's strong conjencture"

"Goldbach's weak conjencture"

What progres has been achieved for more than quarter of a millennium?

"Goldbach's strong conjencture":

• In 1995 professor Olivier Ramaré from Uniwersity in Lille proved that every even number greater than 4 is the sum of at most 6 primes.

• Chinese mathematician Chen Jingrun proved that using the so called "sieve method", that any even number large enough is the sum of two primes or the sum of the prime and semiprime.

• Portugese professor Tomás Oliveira e Silva veryfied the "Goldbach's strong conjencture" to number greater than 15·10¹⁷.

"Goldbach's weak conjencture":

• In 1923, Godfrey Harold Hardy and John Edensor Littlewood showed that, assuming the generalized Riemann hypothesis, the odd Goldbach conjecture is true for all sufficiently large odd numbers.

• In 1937, a Soviet mathematician, Iwan Matveevich Vinogradov , proved that all sufficiently large odd numbers can be expressed as the sum of three primes - it is Vinogradov's theorem.

• In 1939 Brodzin, student of Ivan vinogradov corrected the result and proved that "large enough" in this case means "greater than 3^14348907" - this number has 6846169 digits.

• In 1997 Deshouillers, Effinger, Te Riele i Zinowiew corrected the Hardy's and Littlewood's theorem, proving that generalized Riemann hypothesis is follwed by Goldbach's weak conjencture.

• In 2002 the Brodzin's result was corrected by Liu Ming-Chit and Wang Tian-Ze. They proved that any number greater than n > e³¹⁰⁰ meaning 2·10¹³⁴⁶ matches the criteria of Goldbach's weak conjencture.

We should also mention about the book of a greek writer - Apostolos Doxiadis. He is the author of a well known book that has been translated into 25 languages. It's Polish title is "Killer hypothesis", but the original title and the title of most translations is "Uncle Petros and Goldbach's conjencture". As You can guess the author is describing obsessive attempts of proving the hypothesis by the main protagonist. Despite his knowing that he is doomed to lose, he desperatly fights in search of the solution of the riddle. After publishing of the book by the Faber and Faber publishing house and Bloomsbury Publishing announced that anyone that will solve the problem within two years, will recieve one million dollars. Same prize has been offered by the Clay Institute. If only the money could solve the problem...

Why praticipate in the project?

One can easily say: "If You try to solve the Goldbach's hypothesis you can let it go - most famous of the mathematicians tried it for centuries and achieved nothing" and they would be right in some way. The "strong" version cannot be proved true or false by the modern computers - the problem is the infinite set of numbers. We are in a much better situation when we look at the "weak" version. Liu Ming-Chit and Wang Tian-Ze in 2002 narrowed the set, which in the next few years will be corrected again. It is not an easy task - if it was easy, someone would have solved it already.

As the BOINC System is a public platform, taking advantage of the power of computer time donated by people connected with which to perform the calculations, the results of all the work and source code of the programs should be made public. Each participant should have the possibility to verify the correctness of calculations and so it shall be.

I hope that it didn't bore anyone. I've just like to state the problem in the most simple way - easy to understand for a common user. Mathematicians already know all of this.