Origins of the two Goldbach’s conjectures


Europe, 18th century. While the Western powers were all a flourishing of industries, cultural exchanges and scientific discoveries, the Russian Empire was always one step behind, with an industry that was little more than artisanship, a feudal economy, and a public education that wasn’t even worthy of the name.
But the new Tsar Peter the Great, after touring Europe in search of allies against the Turks who threatened the southern borders, was able to see this abysmal gap with his own eyes, and decided that the time had come for a turning point. He immediately started a great reform campaign, which turned his empire upside down like a sock: he reorganized the state administration in order to make it more efficient, financed the creation of new industries, instituted a new school system, and more. He also founded new commercial hubs, such as Saint Petersburg, built at the mouth of the Neva River on a territory torn from the Swedes after years of war: from a small military fortress, in a swampy area on which no one would have bet a penny, drained and consolidated by the hard work of labourers, master builders and architects, a grandiose and modern city arose, elected as the new capital of the empire.
From the beginning it was clear that the price to be paid would be higher than one could imagine, but progress was a goal to be achieved at all costs, because the alternative would be to see the empire dissolve and get lost in oblivion.
There was also another serious problem to be addressed, which was the real heart of the matter: in the rest of Europe, scientific and technological research was galloping like never before, while in the Russian Empire it had stopped for decades. For this reason, in 1724, the Tsar gave orders to his personal doctor, Lorenz Blumentrost, to found a special institution, to make it a springboard towards the future. It was baptized the Saint Petersburg Academy of Sciences, named after the city where it was built.

Outer façade and inner view of the Kunstkamera, first headquarters of the Saint Petersburg Academy of Sciences; engraving by Grigory Anikiyevich Kachalov, 1740
Outer façade and inner view of the Kunstkamera, first headquarters of the Saint Petersburg Academy of Sciences; engraving by Grigory Anikiyevich Kachalov, 1740

But the Academy was not born to be a university like any other: its purpose was not only education, but also and above all research, because it was necessary to progress, and it should be done quickly.
Many brilliant minds had the honour of participating in the life of the Academy, joining it as professors and researchers; among them there was one of the two protagonists of our story, a Prussian mathematician named Christian Goldbach, who had already made himself known in the scientific community for Specimen methodi ad summas serierum, an important treatise about numerical series. He sent an application for admission to the Academy in 1725, which was not accepted immediately, but earned him the chair of mathematics, as well as a role as a historian; he also had the honour of attending the opening ceremony with the title of Glavnyy uchenyy sekretar ‘Prezidiuma (general scientific secretary of the Presidium), which he also retained later.

A long collaboration

The following year, a serious mourning suddenly struck the Academy: Nicolaus II Bernoulli, professor of mathematics and physics, was struck down by a violent attack of fever. This dramatic event caused a chain reaction: his place was taken by his brother Daniel, who however had to leave his chair of mathematics and mechanics applied to physiology, which then, in turn, lost its teacher. This was a problem, but Daniel already had in mind the perfect name for his successor: a family friend, in whom his father Johann, also a mathematician, had already discovered a great talent for numbers as a child. His name was Leonhard Euler, one of the most important mathematicians who ever existed, who will have a fundamental role in the birth of the Conjecture, as we will see shortly.
Euler accepted the proposal, but initially tried to stall, because in the meantime he was trying to get a chair in physics at the University of Basel, his hometown; in May 1727, when it became clear to him that the attempt was failing, Euler left for Saint Petersburg. Arrived in the city, he took part in the funeral of Catherine I, second wife of Peter the Great and regent of the empire, and then went to a reception, organized by Blumentrost, who became the first president of the Academy: it was here that Euler and Goldbach met for the first time.
Later, there was no lack of opportunities to meet again and also to work together, during lectures, demonstrations of experiments, and more generally in various events in the life of the Academy. It was more like a working relationship between colleagues, but they soon discovered they had something in common: an interest in number theory.
The following year, Goldbach received a proposal that no one would say no to: he was appointed tutor of the young Peter II, who succeeded Catherine as the new tsar, being the only male heir of Peter the Great, who was his grandfather. In order to carry out such an important task, Goldbach moved to Moscow, which was made once again the capital of the empire; therefore, in order to continue having contact with the Academy, he had to use the post, which was the fastest mean of communication existing at the time. He wrote letters to several professors, but the correspondence with Euler became more frequent, since the latter, after reading his paper De terminis generalibus serierum about numerical series, wrote him a letter to share with him some new results he got about the subject; since then the letters, written in German and Latin, began increasingly to focus about approximation, factorization and prime numbers. Many of these letters have come down to us, but not all of them, because some have been lost.
For a while, communication by letters was no longer needed, since, when Peter II died prematurely in 1732, the new empress Anna Ivanovna Romanova moved the capital back to Saint Petersburg, so Goldbach returned to live there and attend the Academy in person, and then see each other with Euler; during this period they often collaborated in various activities of the institution, the most important of which was taking part in a commission, set up by the new president Karl Hermann von Brevern, in order to implement a reform of the finances.
This coexistence continued until 1741, when Euler accepted the invitation of Frederick II, King of Prussia, who offered him a place at the Berlin Academy of Sciences, so, from that moment, the flow of letters was resumed.

Birth of two mysteries

As time passed, the relationship between Goldbach and Euler consolidated, to the point that, when Euler had his first child in 1734, he wanted Goldbach to be his godfather along with the then president of the Academy, Johann Albrecht von Korff. When afterwards in 1738 Euler, following a serious illness that almost cost him his life, lost sight in his right eye, Goldbach took care of talking about it with the president, in order to dispense him from one of his duties, the examination of geographic maps, as part of a project aimed to create a general map of the empire. Both were always very busy, Goldbach because of his duties as Academy secretary, Euler because he was always engaged in solving mathematical problems that had remained unsolved for decades, but they always found the time to carry out their joint research. It involved both insights into the works of other colleagues and new possible theorems; in the letter dated 7 June 1742, for example, Goldbach wrote:

I’d like to risk another conjecture of that kind: any number composed from two primes is the sum of as many prime numbers (including the unit) as one wishes, right down to the sum that consists just of ones(*). For example:
4 = \begin{cases} 1 + 3 \\ 1 + 1 + 2 \\ 1 + 1 + 1 + 1 \end{cases} 5 = \begin{cases} 2 + 3 \\ 1 + 1 + 3 \\ 1 + 1 + 1 + 2 \\ 1 + 1 + 1 + 1 + 1 \end{cases} 6 = \begin{cases} 1 + 5 \\ 1 + 2 + 3 \\ 1 + 1 + 1 + 3 \\ 1 + 1 + 1 + 1 + 2 \\ 1 + 1 + 1 + 1 + 1 + 1 \end{cases}

(*) After reading this through again, I see that the conjecture can be proved quite rigorously for the case n + 1 if it holds for the case n, and if n + 1 can be split into two prime numbers. The proof is very easy.

Auf solche Weise will ich auch eine conjecture hazardiren: dass jede Zahl, welche aus zweyen numeris primis zusammengesetzt ist, ein aggregatum so vieler numerorum primorum sey, als man will (die unitatem mit dazu gerechnet), biss auf die congeriem omnium unitatum (*); zum Exempel:
4 = \begin{cases} 1 + 3 \\ 1 + 1 + 2 \\ 1 + 1 + 1 + 1 \end{cases} 5 = \begin{cases} 2 + 3 \\ 1 + 1 + 3 \\ 1 + 1 + 1 + 2 \\ 1 + 1 + 1 + 1 + 1 \end{cases} 6 = \begin{cases} 1 + 5 \\ 1 + 2 + 3 \\ 1 + 1 + 1 + 3 \\ 1 + 1 + 1 + 1 + 2 \\ 1 + 1 + 1 + 1 + 1 + 1 \end{cases}
(*) Nachdem ich dieses wieder durchgelesen, finde ich, dass sich die conjecture in summo rigore demonstriren lässet in casu n + 1, si successerit in casu n, et n + 1 dividi possit in duos numeros primos. Die demonstration ist sehr leicht.

This additive property of integers has a simple proof, as Goldbach himself states, thanks to a peculiarity. When he talks about prime numbers, Goldbach also includes unity, because at the time 1 was considered prime, while it’s no longer so. So for Goldbach 1 + 3, 1 + 1 + 3, 1 + 5, 1 + 2 + 3, etc., are all sums of primes. Goldbach then states that, if n is the sum of as many primes (including 1) as desired, from a minimum of two onwards, up to a sum of all 1s, and if n + 1 is the sum of two primes (including 1), then also n + 1 is the sum of as many primes (including 1) as desired (up to the sum of all 1s). The “very simple proof” he refers to is as follows:

  • by assumption, n + 1 is the sum of two primes (including 1)
  • by assumption, n is also the sum of two prime numbers (including 1), i.e. n = p + q, so n + 1 = p + q + 1, i.e. n + 1 is the sum of three prime numbers (including 1)
  • by assumption, n is the sum of three prime numbers (including 1), i.e. n = p + q + r, so n + 1 = p + q + r + 1, i.e. n + 1 is the sum of four prime numbers (including 1)

and so on. In general, adding 1 to all the writings of n as the sum of at least two primes gives all the writings of n + 1 as a sum of at least three prime numbers. The writing of n + 1 that could not be obtained with this procedure would be just that as the sum of exactly two prime numbers: therefore it is necessary to assume by hypothesis that such a writing exists.
After this property, Goldbach added another sentence:

And at least it seems to be true that every number greater than 2 is the sum of three prime numbers.

Es scheinet wenigstens, dass eine jede Zahl, die grösser ist als 2, ein aggregatum trium numerorum primorum sey.

Although this property appears to be very different from what we know today as Goldbach’s Conjecture, in reality upon careful analysis it turns out to be equivalent to it, provided that the unity is considered prime also in Goldbach’s Conjecture.

First of all we’ll prove that, if the property stated by Goldbach (henceforth “PG”) holds, then the Conjecture holds, modified so as to consider 1 prime (henceforth “modified Goldbach’s Conjecture” or “MGC”). So let n be an even number greater than 2; we have to show that, if PG is true, then n is the sum of two prime numbers (1 included).
By PG, n + 2 is the sum of three prime numbers (including 1): n + 2 = p + q + r. Since n is even, n + 2 is also even; therefore p, q and r cannot all three be odd, otherwise their sum n + 2 would be odd. Suppose for example that p is even (the same could be done for q and r). Since p is even, it must be p = 2, because 2 is the only even prime (including 1), so n + 2 = 2 + q + r. Simplifying, we obtain that n = q + r, i.e. n is the sum of two prime numbers (1 included).

Now let’s prove the reverse implication: \text{MGC} \Rightarrow \text{PG}. So let n be an integer greater than 2; we have to prove that, if the MGC is true, then n is the sum of three prime numbers (1 included).
We can distinguish two cases:

  • If n is odd, then n - 1 is even; therefore, by the MGC, n - 1 = p + q, where p and q are prime numbers (1 included). Then n = 1 + p + q, i.e. n is the sum of three prime numbers (1 included). Note that, of course, in order to apply the modified Goldbach’s Conjecture it was tacitly assumed that n - 1 \gt 2, i.e. that n \gt 3, but if n = 3 the PG is immediately verified by the equality n = 3 = 1 + 1 + 1, without even needing to assume the MGC to be true.
  • If n is even, then n - 2 is even; therefore, by the MGC, n - 2 = p + q, where p and q are prime numbers (1 included). Then n = 2 + p + q, i.e. n is the sum of three prime numbers (1 included). Note that also in this case, to apply the MGC, a tacit assumption was made, namely that n - 2 \gt 2, i.e. n \gt 4, but if n = 4 the PG is immediately verified by the equality n = 4 = 2 + 1 + 1.

But the investigation did not stop there, because Euler, on the 30th of the same month, sent an answer, in which he proposed some observations:

The fact that any number which can be written as the sum of two primes can also be written as the sum of as many primes as you like can be illustrated and confirmed by an observation which you, sir, communicated to me some time ago, when you stated that every even number can be written as the sum of two prime numbers. Because, if the proposed number n is even, then it is the sum of two primes, and since n - 2 is another sum of two primes, n is also the sum of three, or four and so on. On the other hand, if n is an odd number, it is surely a sum of three primes, since n - 1 is the sum of two primes, and therefore it can also be written as a sum of as many as you like. Indeed, I consider the statement that every even number is the sum of two prime numbers an absolutely certain theorem, although I cannot prove it.

Dass eine jegliche Zahl, welche in zwey numeros primos resolubilis ist, zugleich in quot, quis voluerit, numeros primos zertheilt werden könne, kann aus einer Observation, so Ew. vormals mit mir communicirt haben, dass nehmlich ein jeder numerus par eine summa duorum numerorum primorum sey, illustrirt und confirmirt werden. Denn, ist der numerus propositus n par, so ist er eine summa duorum numerorum primorum, und da n − 2 also eine summa duorum numerorum primorum ist, so ist n auch eine summa trium, und auch quatuor, und so fort. Ist aber n ein numerus impar, so ist derselbe gewiss eine summa trium numerorum primorum, weil n − 1 eine summa duorum ist, und kann folglich auch in quotvis plures resolvirt werden. Dass aber ein jeder numerus par eine summa duorum primorum sey, halte ich für ein ganz gewisses Theorema, ungeachtet ich dasselbe nicht demonstriren kann.

Here Euler clearly quotes what we now know as Goldbach’s Conjecture (or “strong conjecture”), but there are some important things to point out:

  • Euler quotes an observation that you, sir [referring to Goldbach], communicated to me some time ago, when you stated that every even number can be written as the sum of two primes, but we have no transcript of this communication; so either they discussed it in person, or it’s in one of the lost letters. This letter, therefore, is currently the only documented evidence that the strong conjecture was formulated by Goldbach, in the form in which we know it today. On the other hand, as we have seen, Goldbach in the previous letter wrote a statement equivalent to the Conjecture (considering 1 prime), albeit different in form.
  • What has gone down in history as “Goldbach’s weak conjecture” was actually formulated by Euler, when he writes: “if n is an odd number, it is surely the sum of three primes“. Today this property is a real theorem, having been proved by Helfgott in 2013, and it’s stated practically in the same way as Euler, adding only the clarification that n \gt 5, a secondary aspect on which certainly Euler intentionally flew over.

The property stated by Goldbach seen previously, i.e. that every integer greater than 2 is the sum of three prime numbers (1 included), is called in some sources (such as this) “Goldbach’s ternary conjecture”, but be careful, because the same name is also used to refer to the so-called “weak conjecture”, i.e. the property that every odd number greater than 5 is the sum of three prime numbers. Despite the formal similarity, these are two completely different statements: as we have seen, the first is roughly equivalent to the strong conjecture, and therefore has not yet been proved; the weak conjecture instead has been proved (but being called, in the original proof, “ternary conjecture” instead of “weak conjecture”).


Correspondence of Leonhard Euler with Christian Goldbach

4 Replies to “Origins of the two Goldbach’s conjectures”

  1. I think the proof of the strong conjecture should follow like this:
    Given an even number n > 2,
    1) Prove that between 2 and n//2 (integer division) there is at least one prime number A.
    2) Prove that from (n – 2) down to (n//2) a prime number B occupies the same position that some A occupies, in such a way that they are side by side in the two formed rows. A and B will be the terms that adds up to n.

    Example: n = 12.
    A in 2, 3, 4, 5, 6
    B in 10, 9, 8, 7, 6

    So, the terms that adds up to 12 are 5 and 7.

    I’m just a curious person and I would appreciate some kind of reply. I wish to learn. Thank you.

    1. Yes, the proof may work like that, but the main difficulty is to predict the “positions” of prime numbers. The sequence of prime numbers reveals some emerging behaviour, like the Prime number theorem, but there is no regular fine-grained structure.

      However, we adopted similar approaches in our proof strategies. If we replace the concept of prime number by what we call a “space“, which is a number not divisible by some fixed set of consecutive numbers, often prime numbers (e.g. a space can be a number not divisible by 2 and by 3), some predictable fine-grained patterns will appear in the sequence of spaces, and a proof like the one you sketched will become more feasible. Of course, spaces may not be prime numbers, but it’s easy to prove that, if x is not divisible by 2, 3, …, pn, and pn + 1 ≤ x < (pn+1)2, then x is prime (see Properties T.1 and T.2).
      Another similar idea is applied in the proof Goldbach-Bertrand Theorem, which is similar to Goldbach’s Conjecture but much weaker. The proof starts from the existence of a prime between n//2 and n, and then obtains that the other number is coprime with n.

    1. We confirm that Goldbach’s strong Conjecture is still unsolved. Our proof strategies are not complete proofs, they have same important open points which we are working on.

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