Some important results

The statement phrased by Christian Goldbach is a “conjecture”, so, as a matter of principle, it is a hypothesis. This means that it can be:

  • True, that is all even numbers greater than 2 can be expressed as a sum of two prime numbers;
  • False, that is at least one even number greater than two exists, which cannot be written as a sum of two prime numbers.

Both hypotheses are open, and scholars from all over the world are following different paths for finding a solution to what has actually become an enigma. There are in fact different possible scenarios:

  • The Conjecture could be proved to be false simply by finding a counterexample, that is an even number which is not expressible as a sum of two prime numbers. About this question, we know for certain that such a number must be greater than 4 \cdot 10^{18}. In fact, all previous even numbers greater than 2 releaved themselves to be the sum of two prime numbers, as it was verified within the scope of Tomás Oliveira e Silva‘s project.
  • The Conjecture could be proved to be true by a theoretical argument; till now, though, all the many attempts of this kind didn’t end well because of some problems in the argumentation.
  • There is also the possibility that the Conjecture is intrinsically impossible to be proved, or that it’s practically impossible to be refuted. In fact, a hypothetical conterexample may be such a big number to be out of the reach of the means we have for finding it by analyzing even numbers one at a time; on the other hand, also proving the truth of the statement by using the arithmetic we know, based on Peano axioms, might not be possible. In fact, by Gödel’s first Incompleteness Theorem, there are some statements about arithmetic that are not provable, for which, so, it’s impossible to be certain whether they are true or not.

To make matters worse, the Goldbach’s conjecture entails a numerical problem: if it’s true that an even number is a sum of two prime numbers, what are these two numbers? Determine them is rather easy if the starting number is small, but the more it grows, the more finding these two prime numbers becomes in turn “a challenge in the challenge”. Finding a solution for this numerical problem has in turn some theoretical implications: if a formula for computing them was found, and if it would work for any even number, the proofs would follow automatically.
The hunt, so, is more open than ever.

Attempts to decompose even numbers

In order to try to prove that an even number can be written as a sum of two prime numbers, one possibility is to find a way to write it first of all as a sum, and then reduce the characteristics of the related addends, until proving that two of them are enough and they are both prime: in this way, the solution is approached gradually, narrowing things down step by step. The results in this sense are many, and each of them offers a partial solution to the original conjecture, stating that an even number greater than two is certainly a sum of two prime numbers and of other numbers:

  • Viggo Brun, Norwegian, in his book Le crible d’Eratosthène et le théorème de Goldbach published in 1919, proved that every even number greater than two can be written as a sum of two numbers, both of which are the product of no more than 9 prime numbers.
  • The Soviet mathematician Yuri Vladimirovič Linnik has proved, in 1951, that there exist two constants h and k such that all even numbers greater than h are expressible as a sum of two odd prime numbers and of k powers of 2. A successive work, by Roger Heath-Brown and Jan-Christoph Schlage-Puchta, has proved that k is equal to 13.
  • The French mathematician Olivier Ramaré proved, in 1995, that every even number n \ge 4 can be written as the sum of at most 6 prime numbers;
  • The Australian mathematician Terence Tao has proved, in 2012, that every odd number can be written as the sum of at most 5 prime numbers;
  • The previous result has been improved by the Peruvian mathematician Harald Andrés Helfgott, who proved, in 2013, the “weak” Goldbach’s conjecture, which states that every odd number greater than 5 can be expressed as a sum of three prime numbers. The direct consequence is that every even number greater than 2 can be written as a sum of 4 prime numbers.

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Chen and Yamada’s theorems

Commemorative stamp dedicated to Chen Jingrun

The result which get most close to the proof of the Goldbach’s conjecture was obtained by a Chinese mathematician, Chen Jingrun. Born in 1933 in Fujian, a district of south-eastern China, lived his youth while the Chinese empire was becoming the People’s Republic we know today. Always science enthustiast, when his math teacher at high school talked to the students about Goldbach’s conjecture, our Chen decided to try to prove it. He dedicated several years at studying it until, in 1966, using Sieve theory as a basis for his argument, wrote a paper, On the representation of a large even integer as the sum of a prime and a product of at most two primes, in which he described the most important result he had achieved, which the rest of the world would later refer to as Chen’s theorem:

Every sufficiently large even number is a sum of two prime numbers, or of a prime and a semiprime (that is the product of two prime numbers).

In arithmetical terms, the theorem can be rephrased in this manner:

There exists an even number k > 0 such that, for every even number n > k, we have that n = a + bc, where a and b are prime numbers, and c is 1 or a prime number.

The original theorem states that k exists, but it does not indicates its value. An important result in this sense is due to a Japanese mathematician, Tomohiro Yamada, of the Osaka university, who proved that k = ee36, where e is the known Napier’s constant. Una curiosità: e is also called Euler’s number… the same Euler who Christian Goldbach talked to about his conjecture for the first time.

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Study of the counterexample set

In order to understand if Goldbach’s conjecture is true, two methods are usually followed: a proof is attempted, or a refutation. An alternate tactic, besides choosing a method or the other, consists in adopting a more direct approach: trying to quantify, to the total, the “counterexamples”, i.e. the even numbers which cannot be written as a sum of two prime numbers. In this way, depending on the result, a conclusion can be directly reached: the conjecture is true if there aren’t counterexamples, otherwise, if almost one exists, it’s false.
Attempts in this way are of different types:

  • Three mathematicians, Soviet Nikolai Chudakov, Dutch Johannes van der Corput and German Theodor Estermann, between 1937 and 1938, have proved that the fraction of numbers satisfying the conjecture tends to 1, that is almost even numbers can be written as the sum of two primes: this result, then, implies that the even numbers not respecting the conjecture, if existing, are very few. The three proofs, reached independently by each scholar, are based on the studies by another Soviet mathematician, Ivan Matveevič Vinogradov.
  • Previous result has been refined later in 1975, by Hugh Montgomery from USA, and Robert Charles “Bob” Vaughan from UK, who determined that, given N, the quantity of even numbers less than N not satisfying the conjecture is less than CN1-c, where c and C are two constants greater then zero.
  • Wen Chao Lu from China proved the number of counterexamples less than a sufficiently big number x is less x0,879 (up to a constant factor), improving a similar result obtained by already mentioned Chen Jingrun. This result is particularly important because it may be used for obtaining an alternative proof of weak Goldbach’s Conjecture, as it has been noted by our reader Ultima (Ultima – Almost all odd numbers can be understood as the sum of three prime numbers (alternative proof)).

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The probabilistic approach

A further approach to understanding Goldbach’s Conjecture, not completely rigorous but worthy of mention, is based on the calculus of probabilities, through which some intuitive justifications of the Conjecture can be formalized. For example, in the article A Statistician’s Approach to Goldbach’s Conjecture, Neil Sheldon concludes that the probability that Goldbach’s Conjecture is false is approximately 10^{-} (try to imagine a probability value with 150 billion zeros after the decimal point!). The proof is based on the assumption that, if an even number 2n is the sum of two integers p and q, the probability that p is prime has no influence on the probability that q is prime, i.e. the primality of p and the primality of q are independent events. This assumption is not proved, so Sheldon’s proof cannot be considered complete; however it is interesting that the probability value finally obtained is so low.

Tomàs Oliveira e Silva’s project

Along with the theoretical way, Goldbach’s conjecture can be also studied by numerical methods, trying to find the two prime numbers which every even number is the sum of. In this way, we can study how these numbers behave, if they follow some rule, and so on.

Professor Tomás Oliveira e Silva, of Aveiro University in Portugal, followed exactly this way: he started a project, ended in 2013, for realizing an authomatic method for writing as a sum of two prime numbers all even numbers up to 4 x 10 18. Resources were the main problem: as the number to be scomposed increased, the necessary computations required more and more time and computing power from computers. In order to be successful in completing the work in reasonable time, above all in the final part in which the numbers involved were very big, the contribution of an Italian, Silvio Pardi, of the Naples location of the National Institute of Nuclear Phisics, was essential.
In this way, it was possible to state that Goldbach’s conjecture is certainly true for all even numbers up to 4,000,000,000,000,000,000, four billions of billions. It’s a very big number, not enough by itself for proving that the conjecture is always true, however still representing a certain proof that, if an even number not satisfying the conjecture exists, it would be surely bigger than that.

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Based on the presented results, we can conclude that, so far, we know that some even numbers are the sum of two primes, while for others we have only partial information. The current situation can be summarized in the following scheme (not in scale):

4 A – Oliveira e Silva 4 · 1018  
  8 B – Helfgott        
  ee36 C – Chen-Yamada        
  • Section A contains all even numbers which are surely the sum of two primes; this result is due to the work by Oliveira e Silva.
  • Section B contains even numbers which are the sum of 4 prime numbers; this result is a consequence of the weak Goldbach’s Conjecture, proved by Helfgott.
  • Section C contains even numbers which are the sum of two prime numbers or of a prime and a semiprime; this result is Chen’s Theorem, combined with Yamada’s further work.

This is what we know so far, but the research is still in progress, so the situation could change. In particular:

  • Computers become more and more powerful, so it will be possible to decompose more and more even numbers into pairs of prime numbers, therefore the 4 · 1018 threshold is destined to become outdated. It is not excluded that, sooner or later, a counterexample will be found, so this path might definitively close the matter.
  • Compared with Goldbach’s Conjecture, Chen’s Theorem is the most similar theoretical result which we have reached so far, but, sooner or later, it may be improved. For example, the minimum limit ee36 found by Yamada could be reduced. In particular, if this minimum boundary was brought below the maximum boundary of section A, Chen’s Theorem would be valid, by extension, for all even numbers greater than 2 (because it would be already known that the ones below the minimum threshold respect it), thus becoming even more similar to Goldbach’s Conjecture.
  • The same discourse of Chen’s Theorem could be applied, in the future, to Goldbach’s Conjecture itself: if it was possible to prove that sufficiently large even numbers are the sum of two prime numbers, it would be an excellent starting point. The next steps would be to establish a minimum threshold, as Yamada did for Chen’s Theorem, and then to try to progressively lower that threshold, until it falls back into section A, thus definitively proving the Conjecture. In this case, Chen’s Theorem would be definitively overcome, because the set of even numbers expressible only as the sum of a prime and a semiprime would turn out to be empty. In other words, every even number expressible as the sum of a prime and a semiprime, would also admit another expression as the sum of two prime numbers, therefore the second possibility considered in Chen’s Theorem would turn out to be superfluous.

4 Replies to “Some important results”

  1. To me it seems important that Goldbach’s Conjecture can be reformulated as a statement about the distribution of the odd semiprimes in relation to the perfect squares – such that for squares of even numbers greater 2 there should exist a lower odd square such that their difference is an odd semiprime, and such that for squares of odd numbers greater 3 there should exist a lower even square such that their difference is also an odd semiprime – because then, adding the non-trivial factors of the semiprimes should yield a “perfectly dense” set of even numbers greater 4.

    1. Yes, Goldbach’s Conjecture can be reformulated as follows:

      2n = p + q =>
      4n2 = p2 + 2pq + q2 =>
      n2 = (p2 + 2pq + q2)/4 =>
      n2 = (p2 – 2pq + 4pq + q2)/4 =>
      n2 = (p2 – 2pq + q2)/4 + pq =>
      n2 = [(p – q)/2]2 + pq

      Is this reformulation important? The answer is debatable, but it’s a fact that we made just a few simple algebraic passages. So, though the result looks very different from the original statement, the path travelled from the former to the latter is very short. We think that whoever is able to prove the form “n2 = [(p – q)/2]2 + pq” would also be able to prove the form “2n = p + q”, and vice versa.

  2. Is there any effort to achieve partial proofs regarding infinite subsets of natural numbers ?

    For instance, even subprimes are by definition the sum of two primes. What about powers of two ? They certainly cannot be expressed as the sum of two composite numbers, unless those composites are coprime. How about squares of even numbers in general ? Primorials ?

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