Some mathematical statements are deceptively simple. Goldbach's conjecture is one of those statements. The story starts on the 7th of June 1742 when mathematician Christian Goldbach wrote a letter to his friend Leonhard Euler that he couldn't prove the following conjecture: "Every even number greater than two is the sum of two primes." Euler wrote back that he was certain the statement holds true but was unable to verify it mathematically. Nearly three centuries later, no one has been able to prove the statement.
There is also something appealing about unsolved mathematical problems. Problems like the Collatz Conjecture, the Twin Prime Conjecture, and P versus NP are all easy to state, yet very difficult to prove.
The simplicity of these problems often makes me curious: “What if there is a simple idea everyone else missed? Maybe I can prove it?” That curiosity, combined with the accessibility of mathematics nowadays, makes it that people want to try to prove these problems.
While Goldbach may not be proven, many mathematicians still believe the conjecture to be true. One reason is computational evidence. The conjecture has been verified for all even numbers up to 4x1018. Another important reason comes from probability theory. The Prime Number Theorem states that near a large number x, prime numbers occur approximately with the probability 1/ln(x). Now consider an even number 2n = p + q, where p and q are prime. In how many ways can such a representation occur? There are roughly n candidate pairs where each candidate pair has a probability of 1/ln(n)2. So, the expected Goldbach representations are roughly n/ln(n)2, which increases as n increases. Heuristically, this means that large numbers typically would have more Goldbach decompositions. However, a heuristic argument is not a proof and converting such an argument into a mathematical proof is extremely difficult.
As I was interested in the problem myself, I stumbled upon this proposed proof published on the HAL archive. The proof attempts to derive Goldbach using Dirichlet’s theorem but contains some logical errors. Goldbach's conjecture can be formulated as follows:
Fixing one prime p we can write q = 2n − p and consider the following sequence: 2n − 3, 2n − 5, 2n − 7, . . . The proof aims to show that one of these numbers must always be prime. In his proof the author rewrites the sequence into an alleged arithmetic progression and invokes Dirichlet's theorem. However, this is precisely where the reasoning goes wrong.
Dirichlet's Theorem is stated as follows. An arithmetic progression of the form: a, a + b, a + 2b, . . . where the greatest common divisor between a and b is 1, contains infinitely many primes. For example, take a = 2 and b = 3. The sequence 2, 5, 8, 11, . . . then has infinitely many prime numbers. But this does not mean that there is a prime in every position.
Goldbach's conjecture requires:
This means that for all n we must prove that there exists a prime number p such that 2n − p is also prime.
Dirichlet's theorem gives us the following:
For a fixed prime p not equal to 2 Dirichlet’s theorem implies that the arithmetic progression 4 − p, 6 − p, 8 − p, . . . contains infinitely many primes. But this is not what Goldbach’s conjecture requires. The order of p and n has been swapped, which makes the logic flawed. It would be the same logical mistake as saying that every person having a favourite colour implies that there exists a colour which is everyone's favourite.
The second flaw in the proof is that the proof appears to treat the sequence 2n − 3, 2n − 5, 2n − 7, . . . as an arithmetic progression, while it is not. If the sequence were an arithmetic progression, the next number would be 2n − 9, but 9 is not a prime and the number doesn't belong in the sequence. This matters because Dirichlet’s theorem only applies to genuine arithmetic progressions. We can conclude that this proof of Goldbach is invalid.
The accessibility of online publishing has made mathematics far more open. That openness is valuable but also increases the importance of careful verification. A convincing looking proof can have subtle errors, which makes careful analysis essential when reading mathematical claims online. As we’ve seen Goldbach’s conjecture remains unsolved. It does so not because mathematicians have overlooked simple arguments, but because the problem is profoundly difficult. And a simple problem can be very hard to prove.