Goldbach and his conjecture: when pairs become prime

The year was 1742. Christian Goldbach , a Prussian mathematician with very serious concerns, wrote to his colleague Leonhard Euler (no more and no less than the inventor of the number e ) and told him the following:

“I believe that every even number greater than 2 can be written as the sum of two primes.”

Just like that, without any anesthesia. No jokes. Euler didn't laugh, nor did he refute it. He found it interesting, and since then the conjecture has lived a long life... unsolved.

Imagine someone tells you that all even numbers greater than 2 can be written as the sum of two prime numbers. Sounds simple, right? So simple that it seems like it can't possibly be true, and you're able to quickly mentally come up with a counterexample that proves it's not true.

But wait a minute.

We're talking about even numbers , those that, except for 2, aren't prime. They're like "ordinary" numbers, those that are always divided equally. And then there are prime numbers , just the opposite: the solitary ones, those that can't be divided by anything other than themselves and 1.

And suddenly someone says that all those even numbers, without exception, can be formed by adding two of those lone numbers together… it's like saying that any pop song can be composed by mixing just two jazz notes. 🤯

It's one of those moments where your brain fries. Because you try it with the 4, with the 6, with the 10, with the 100... and it works. And you keep going, and it works. And you wonder: how the hell is this possible?

Well, that same feeling of "this can't be right, but it seems so" is what has plagued entire generations of mathematicians. Ladies and gentlemen: presenting Goldbach's conjecture , the most elegant (and maddening) puzzle in number theory.

A little bit of theory

Goldbach's conjecture states that:

"Every even number greater than 2 can be written as the sum of 2 prime numbers."

Above only two , not even a complex sequence of primes.

Starting with the first value of the guess, we have the following:

• 4 = 2 + 2
• 6 = 3 + 3
• 8 = 3 + 5
• 10 = 5 + 5 (also valid 3 + 7)
• 12 = 5 + 7
• 14 = 3 + 11
• 16 = 3 + 13 (also 5 + 11)
• 18 = 5 + 13 (or 7 + 11)
• 20 = 3 + 17 (or 7 + 13)
• 22 = 3 + 19 (or 5 + 17)
• ...
• 100 = 47 + 53
• 1000 = 3 + 97
• 25426 = 10979 + 14447

And so on, as far as calculators dare.

Why is it a conjecture and not a theorem?

Here comes the frustrating part for mathematicians: all pairs up to 4 × 10¹⁸ have been proven by computer to satisfy the conjecture. No one has found a single counterexample. But… there is no general mathematical proof .

And in mathematics, you know how it goes: “we have proven it to infinity” is no good unless you prove it to all possible infinities .

This is so important because prime numbers are the raw material of the numerical universe. They're like the atoms of mathematics: everything is built from them, but understanding how they behave is like trying to predict a cat's mood.

That's why they've always been the focus of research, demonstrations, theories, and technological advances. And Goldbach's conjecture touches on the heart of that chaos: the distribution of prime numbers . Solving it would be like bringing order to the purest disorder. An epic feat.

Goldbach's ternary conjecture

As if Goldbach wasn't content with leaving us with one of the most resilient conjectures in history, he left us another one. A bonus, an extra, a "just in case": Goldbach's weak conjecture , also known as the ternary conjecture .

This version says that:

Every odd number greater than 5 can be written as the sum of three prime numbers.

For example:

• 7 = 2 + 2 + 3
• 9 = 2 + 2 + 5
• 15 = 3 + 5 + 7
• 33 = 3 + 11 + 19

And yes, it looks less elegant than the original, more “handy,” but… this one really caught on .

In 2013 , Peruvian mathematician Harald Helfgott earned his place in the books by proving it with a combination of analytical number theory and brutal computational verification. It was a colossal breakthrough that closed a chapter unfinished since the 18th century.

It is a kind of “training version” of the strong conjecture (the original one), but the fact that it was proven was a very important advance in the field.

Why is it still called a “conjecture” if it has already been proven?

Because the original name doesn't change after it's been proven . In mathematics, a conjecture is a proposition that is believed to be true but hasn't yet been proven . When it's finally proven, it becomes a theorem , but it's often still called a conjecture out of historical tradition or fame.

Classic example: Fermat's conjecture: after centuries of unsolved problems, it was finally proved by Andrew Wiles in 1994. And what do we still call it? Well, that's right, Fermat's conjecture , even though it's already a theorem .

The same thing happens with Goldbach's weak conjecture . It's been known as a conjecture since 1742, and although Harald Helfgott proved it in 2013 , the mathematical community still refers to it by that name. Changing it to "Goldbach's ternary theorem" would be technically correct, but... nobody does it.

Conclusion: A simple idea, an eternal mystery

The fascinating thing about Goldbach's conjecture isn't just that no one has been able to prove it, but that it's so devilishly simple . You don't need to know derivatives, matrices, or integrals to understand it. You just need to know how to add. And yet, there it is: resisting the greatest geniuses for almost three centuries.

It's the kind of problem that reminds you that in mathematics, the simple isn't always easy. Sometimes the most innocent is the most profound. And as long as no one can prove (or disprove) it, it will remain there, hovering over our heads as a reminder of how little we still understand about the numerical universe.

Maybe tomorrow someone will find definitive proof. Or maybe another 300 years will pass. But in the meantime, every even number greater than 2 continues to play its part... as if Goldbach, from some corner of infinity, were winking at us.