A legendary letter was sealed more than three centuries ago. What was in it was a sort of puzzle, a challenge so simple that any elementary student could understand. However, it is such a complex problem to prove that the greatest minds in history have failed trying. Between the lines of that letter, one said the following:
That is the whole problem. The whole numbers are those with which we learned to count: the one, the two, the three … Without frills. A prime number is all that greater than one and you can only divide it between yourself and the unit without decimals: two, three, five, seven, eleven … That and adding is the only thing you need to know To understand the problem, there is no more. This is the conjecture that Christian Goldbach proposed to Leonhard Euler in 1742, the unbeatable beast that has defeated every great mind that has dared to face her. However, I am sure that a small part of you has been tempted to test it. So why not try?
Let’s try the eight. We can do this by adding three and five, so it is fulfilled. And the fourteenth? It would be three plus eleven, and twenty one is two and nineteen. I assure you that you can try with everyone you want, because you will not find a single case in which the conjecture is not fulfilled. And we know this because some machines have tried before you. Computers have checked the first four trillion numbers, a four followed by eighteen zeros. It might seem that this is enough to prove that the conjecture is true, after all, four trillion numbers supporting it are many numbers. However, mathematicians do not get that. You have to be sure that there is not a lost number out there that is not met, because, as huge as it may seem, four trillion are left at nothing compared to the infinite ocean of figures out there.
Mathematicians need to be totally and absolutely sure, everything else is pure opinion and only with opinions do not do mathematics. So, there are mostly two ways to be satisfied. The first is the simplest, the one we all try when we are taught the problem: find a counterexample, take the opposite, find a case that is not met so that we can reject the conjecture. That is what we have tried, but as we have seen, it does not seem very fruitful … although, wait. And the two? It is an integer and even number, but it is not the sum of two cousins because it can only be formed by adding two ones, and the number one is not a cousin. For the 2 Goldbach is not fulfilled! Is it possible that we have resolved Goldbach’s conjecture?
I’m afraid not, because this apparent lack of confidence is a historical curiosity. In the 18th century, the 1 was still considered a prime number, so Goldbach could express the two as “one plus one” and remain so calm. In fact, excluding one from the list of prime numbers had to update the conjecture. The modern version would be something like this:That every integer even greater than two is the sum of two prime numbers seems to me to be a completely true theorem, but I can’t prove it“So let’s forget number two and worry about the rest.
As we have seen, a counterexample has not yet been found, and although by definition there are still infinite numbers to prove, mathematicians suspect that no matter how much they search they will not find a single case in which Goldbach is not fulfilled. They argue that, as the numbers grow, the ways in which they can express themselves also increase. For example: ten can be done by adding five to itself or three to seven. One hundred, for example, can be constructed in six different ways, and from what we have seen, the number of possible combinations skyrockets as we study larger numbers. In fact, there is a very visual way to see it: Goldbach’s comet.
It is a graph whose horizontal axis represents the number to study and in the vertical we find how many different forms we can decompose. The key is that the comet does not stop growing and it becomes difficult to think that at some point it will plummet by marking zero on the vertical axis, indicating an even number that cannot be constructed by adding two cousins. However, this is still not enough evidence. It is only a suspicion, but nothing assures us that there is an exception among all the numbers that we cannot check by hand, so we have to change our strategy.
Luckily there are other ways to prove a conjecture: finding a general argument applicable to all numbers and showing that, if Goldbach were false, mathematics would have to contradict themselves. Something like looking for a nonsense, showing that we are facing something impossible. A classic example of this strategy is Euclid’s theorem, which posed the existence of infinity of prime numbers. His way of demonstrating it was as follows:
Let’s take a few prime numbers and multiply them together, for example: 2x3x7. The result is never a prime number, because it is divisible among all the previous ones giving, in this case, 42. Now we must add a 1 and bingo! We have already achieved a prime number, 43, because adding 1 stops being divisible among the cousins we have used to build it. However, this is not always the case, for example: 3×5 + 1 gives 16, which is not a cousin. And here is the final trick, because, if the result is not a cousin and at the same time it is not divisible among the cousins we have used to build it, it means that its divisor is a new prime number, in this case the two: 2x2x2x2. Following these steps we can build as many prime numbers as we want, demonstrating that there are infinity of them.
This type of reasoning is what mathematicians look for, ways to prove or deny conjectures beyond any doubt avoiding logical inconsistencies. However, these approaches have also failed to resolve Goldbach’s conjecture. Nothing has turned out, and although from time to time someone appears saying they have tried it, they are always false alarms that show nothing. However, you may have heard that some years ago a mathematician resolved Goldbach’s blissful conjecture, and it is “true,” except that it was Goldbach’s weak conjecture, something quite different.
In 2013 Harald Andrés Helfgott managed to prove Goldbach’s weak conjecture, which reads as follows: “Any odd integer greater than 5 can be expressed as the sum of three prime numbers”For example: 15 is the sum of 3, 5 and 7. Helfgott’s strategy was somewhat intermediate to the two we have proposed.
On the one hand, the computers had already calculated that the weak conjecture was fulfilled for all odd integers between 5 and 8875×10 ^ 30 (representing approximately 9 followed by thirty zeros). On the other hand, some great experts in number theory had shown that, from a sufficiently large number, all of the following had to fulfill the conjecture. The demonstration of the latter is very long for this article, but the important thing is to be clear that this “large enough number” has been reduced with better evidence over the years and that when Helfgot faced the conjecture, he was in 2×10 ^ 1346 (a 2 followed by one thousand three hundred forty-six zeros).
That means we already knew that the largest and smallest numbers met the conjecture, but between them there was a huge chasm and their mission was to reduce it. He could have looked for better computing methods so that computers were able to calculate even more numbers by brute force. However, 8875×10 ^ 30 was already something huge (to give you an idea, it is estimated that there are 1×10 ^ 80 protons in the universe) and as much as they improved, the machines could not reach 2×10 ^ 1346.
So Helfgot decided to face it in another way and try to reduce that number “large enough” to make it smaller than 8875×10 ^ 30 so that the conjecture was demonstrated for all integers through one or another method. And he succeeded, he managed to lower it to overlap it with computer calculations. Goldbach’s weak guess was true.
Meanwhile, the strong conjecture remains undefeated and has not advanced much in recent decades. A drought that is largely due to its complexity, but to which many other things may have contributed. Not a few professionals believe that focusing their research on the Goldbach conjecture is almost a condemnation of failure. Many of those who work in it do it timidly, in their free time and without being able to devote the time they deserve. This perception is so popular that it is treated even in one of the most famous mathematical novels: Uncle Petros and the Goldbach conjecture, from Apostolos Doxiadis.
At this step the question is obvious: Will Goldbach’s conjecture ever be resolved? Will it be true and will be consolidated as a theorem? Or in an unexpected turn of events will it have to be ruled out? It only remains to wait while Goldbach’s old statement echoes in our ears, like a siren song that makes anyone who agrees to dance with her run aground against the cliffs.
DON’T KEEP IT UP
- The conjecture has not yet been proven. It is true that in 2013 Helfgot demonstrated Goldbach’s weak conjecture, but when we refer to “Goldbach’s conjecture” we are talking about the strong one.
- The Fundamental Theorem of Arithmetic implies that 1 cannot be a cousin by saying that: any integer greater than 1 can be written as a single product of prime numbers. If we accept 1 as a cousin there would be infinite ways to break down each number (10 would be 2×5, but also 2x5x1, 2x5x1x1 and so on). There are more reasons, but this is possibly the easiest to understand.
- Christian Goldbach, “Letter to Leonhard Euler” (Moscow, June 7, 1742)
- Leonard Eugene Dickson “History of the Theory of Numbers, Volume I: Divisibility and Primality” Dover Books on Mathematics (2005)
- Harald Andrés Helfgott “Major arcs for Goldbach’s problem” arXiv (2013)
- Tomás Oliveira e Silva, Siegfried Herzog & amp; Silvio Pardi, “Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4×10 ^ 18” Mathematics of Computation 83: 2033-2060 (2013)