The Evolution of Goldbach's Conjecture: From a 1742 Letter to a Claimed AI Proof in 2026

In June 1742, Prussian diplomat and amateur mathematician Christian Goldbach jotted down a seemingly simple arithmetic observation at the end of a letter to the great mathematician Leonhard Euler.

Euler replied that this observation could be distilled into a single statement:

Every even integer greater than 2 can be expressed as the sum of two primes.

Euler called it a “completely certain theorem,” yet admitted he could not provide a proof. While even an elementary school student can check examples by hand, proving that it holds for every even integer greater than 2 remains an open question in number theory.

Why Is Adding Primes So Difficult?

A core difficulty of this problem is how primes—defined multiplicatively—can be made to satisfy an additive constraint.

The definition of a prime is fundamentally about divisibility: a prime is divisible only by 1 and itself. The Fundamental Theorem of Arithmetic says every integer decomposes uniquely into a product of primes.

Yet Goldbach’s conjecture poses an additive question: what numbers can be formed by adding two primes together?

Knowing that 97 is prime does not guarantee that $100 - 97 = 3$ is also prime. For every new even number, one must verify afresh whether what is left after subtracting a prime is itself prime.

From Finding Primes to Counting Partitions

Viewed individually, prime numbers appear almost randomly and unpredictably; zoomed out, however, they obey remarkably stable statistical laws.

One way mathematicians study this problem is to count how many ways each even number can be written as the sum of two primes. For example, 10 has two such partitions: ${3 + 7}$ and ${5 + 5}$ (reversing the order does not count as a separate partition). What is being counted is the number of partitions that satisfy the prime requirement.

If an even number greater than 2 has no pair of primes that add up to it, it is a counterexample to the conjecture. Conversely, proving that every even number greater than 2 has at least one pair of primes that add up to it would prove Goldbach’s conjecture. Counting partitions reformulates the same problem in a way that lets mathematicians use statistical patterns in the distribution of primes; proving that every even number greater than 2 can be written as the sum of two primes remains the challenge.

How the Strong and Weak Conjectures Relate

Historically, research has centered around two formulations:

• Strong (Binary) Goldbach Conjecture: Every even integer greater than 2 can be written as the sum of two primes (e.g., ${4 = 2 + 2}$, ${10 = 3 + 7}$, ${100 = 3 + 97}$).
• Weak (Ternary) Goldbach Conjecture: Every odd integer greater than 5 can be written as the sum of three primes (e.g., ${9 = 3 + 3 + 3}$, ${21 = 5 + 5 + 11}$).

The logical relationship between them is straightforward: if the strong conjecture holds, the weak conjecture follows immediately (subtracting 3 from any odd number greater than 5 yields an even number, whose two prime summands complete the triple). Proving the weak conjecture, however, does not imply the strong one.

The Circle Method: How the Ternary Weak Conjecture Was Solved

In the 1920s, British mathematicians G. H. Hardy and J. E. Littlewood pioneered the circle method.

How Does the Circle Method Guarantee at Least One Prime Partition?

The circle method studies the number of prime partitions discussed earlier. It translates the condition that primes add up to a given integer into mathematical expressions that can be analyzed, using patterns in the distribution of primes to estimate how many combinations meet that condition.

An estimate is not an exact answer, however, so the error must also be controlled. If the estimated number of partitions remains positive even after subtracting the largest possible downward error, at least one valid combination of primes must exist. A rigorous bound on the error is needed to rule out the possibility that the actual count is zero.

In the circle method, the part that provides the main estimate is called the “major arcs”; another part whose contribution must be controlled is called the “minor arcs.” The key to following this history is whether mathematicians can control the error well enough to guarantee that a prime partition exists.

For sums of three primes, mathematicians can do this, at least when the target odd integer is sufficiently large. For sums of two primes, current methods cannot provide such a guarantee for every even integer greater than 2.

From “Sufficiently Large” to All Odd Integers

In 1937, Soviet mathematician I. M. Vinogradov improved the error estimates in the circle method, proving unconditionally (without relying on unproven hypotheses) that all sufficiently large odd integers can be written as the sum of three primes.

That said, “sufficiently large” in the weak conjecture left a loose end: the proof only covered odd numbers above a massive theoretical threshold. Early estimates placed this threshold as high as $10$ raised to the power of millions—a figure far exceeding the total number of atoms in the observable universe (roughly $10^{80}$), rendering brute-force computer verification impossible.

It was not until 2013 that Peruvian mathematician Harald Helfgott sharpened the analytical bounds and, collaborating with David Platt, pushed computational verification well beyond them: those refined bounds lowered the theoretical threshold to $10^{27}$, while rigorous computer calculations verified all odd numbers up to $8.875 \times 10^{30}$.

 Ternary Goldbach Resolution

          5
          |
          | Computer check
          |
        10^27
          |
          | Both methods
          |
  8.875 x 10^30
          |
          | Analytical proof
          v
       Infinity

With the two regimes cleanly overlapping, the weak Goldbach conjecture was fully proven.

Sieve Limits and Chen’s Theorem: The Intractable Parity Problem

The other primary route to tackling the strong conjecture is sieve theory, which traces its roots back to ancient Greece.

Relaxing the Requirement: Numbers with Few Prime Factors

First, consider how prime factors are counted: ${77 = 7 \times 11}$, so 77 has two prime factors; 23 is itself prime and has just one. Repeated factors count too: ${9 = 3 \times 3}$ also has two.

Sieve methods progressively eliminate numbers divisible by small primes. In studying Goldbach’s conjecture, mathematicians use such methods to seek a result that is easier to prove: can every sufficiently large even integer be written as the sum of two numbers, each with very few prime factors? If both summands can be restricted to a single prime factor—meaning both are prime—the original conjecture’s requirement is met.

Results of this kind are often written as ”$r + s$.” Here, $r$ and $s$ record the maximum number of prime factors allowed in each summand, not the values being added. For example, "$1 + 2$" means that one summand is prime, while the other may be either prime or the product of two primes.

Along this route, mathematicians gradually reduced the number of prime factors allowed:

• 1919 (Viggo Brun): Proved ”$9 + 9$,” allowing at most 9 prime factors in each summand.
• 1947 (Alfréd Rényi): Proved ”$1 + c$,” guaranteeing that one summand is prime while the other’s number of prime factors is bounded by a fixed constant $c$.
• 1966/1973 (Chen Jingrun): Published Chen’s theorem, proving ”$1 + 2$.” For example, ${100 = 23 + 77}$, where 23 is prime and 77 is the product of two primes. The theorem guarantees that every sufficiently large even integer has a partition meeting these conditions.

Why Is the Final Step Still Out of Reach?

Chen’s theorem allows the second summand to have two prime factors; the original conjecture requires it to be prime, with just one. Moving from "$1 + 2$" to "$1 + 1$" may seem like a small step, but it runs into the parity problem, identified by Atle Selberg.

Here, “parity” refers to whether the number of prime factors is odd or even: a prime has 1 prime factor, an odd count; the product of two primes has 2, an even count. This is a separate question from whether the integer itself is odd or even.

For a specific number such as 77, we can of course determine that it has two prime factors. The difficulty arises when proving that suitable partitions exist for all sufficiently large even integers: traditional sieve methods rely on statistical information about divisibility by small primes across a collection of numbers. That information alone cannot guarantee a partition in which the second summand is prime rather than the product of two primes.

Thus, Chen’s theorem narrows the possibilities to one or two prime factors, but guaranteeing that both summands are prime requires information or methods beyond traditional sieves.

Why Statistics and Computation Support the Strong Conjecture

Although the strong conjecture has no complete proof, statistical models, computational verification, and bounds on the number of possible counterexamples all provide evidence in its favor.

More Prime Partitions, and No Counterexample Found by Computers

Hardy and Littlewood developed a probabilistic model predicting that larger even integers generally have more ways to be written as the sum of two primes. This supports the strong conjecture: as the expected number of prime pairs grows, finding none seems increasingly unlikely. But the model’s predictions still cannot guarantee that every even integer greater than 2 satisfies the conjecture.

Actual calculations also reveal many partitions: 10,000 has 127, and 100,000 has 810 (reversing the order of the two primes does not count separately). Plotting many even integers against their numbers of prime partitions produces a comet-like band known as Goldbach’s comet.

Computer verification gives a definite answer within a finite range: every even integer greater than 2 and no greater than $4 \times 10^{18}$ can be written as the sum of two primes. However, there are infinitely many even integers. Checking any finite range, however large, cannot guarantee that no counterexample exists beyond it.

Why Does “Almost All” Still Not Mean “All”?

Beyond models and computer verification, the circle method has produced a rigorous theoretical result: almost all even integers satisfy the conjecture. This means that as the range under consideration grows, the proportion of even integers that violate the conjecture tends to zero.

But a proportion tending to zero does not mean there are no exceptions. For example, the squares 1, 4, 9, 16, and so on make up an ever-smaller fraction of the positive integers, yet there are infinitely many of them. Likewise, the result that almost all even integers satisfy Goldbach’s conjecture neither establishes that counterexamples exist nor rules them out.

Research further limits how many counterexamples there could be: for sufficiently large $X$, the number of counterexamples below $X$ is less than $X^{0.72}$. This bound shows that even if counterexamples exist, they must form an ever-smaller proportion. Proving Goldbach’s conjecture still requires moving from “possibly very few” to “none at all.”

September 2026: What Proposition Did the AI Actually Prove?

In September 2026, headlines proclaimed a major AI breakthrough on Goldbach’s conjecture.

The claim stemmed from a Lean formalization posted by the anonymous account Captain Sude—Lean is a system that verifies mathematical logic line by line as computer code. The account’s owner says the proof was produced by GPT-6 Astra, though as of October 2026, OpenAI has not confirmed that attribution.

What Did the Result Really Establish?

This work relaxes the original conjecture’s requirement: every even integer greater than 2 can be written as the sum of two positive integers, each with an odd number of prime factors. Repeated prime factors still count separately.

An odd number of prime factors can mean 1, 3, 5, and so on. A prime has just 1 prime factor, so it qualifies. But ${8 = 2 \times 2 \times 2}$ has 3 and also qualifies, even though 8 is composite.

Comparing two partitions of the same even integer, 16, makes the difference clear:

• ${16 = 3 + 13}$: both summands are prime, meeting both the original conjecture’s requirement and the variant’s condition.
• ${16 = 8 + 8}$: each summand has 3 prime factors, meeting the variant’s condition without being a sum of two primes.

This classification by whether the number of prime factors is odd or even can be expressed using the Liouville function: an odd count is assigned $-1$, and an even count $+1$. The proposition proved here says that every even integer greater than 2 can be written as the sum of two positive integers assigned $-1$.

Why Does Proving the Variant Leave the Original Conjecture Unsolved?

The original conjecture requires both summands to be prime; the variant allows composite numbers such as 8. Proving that every even integer has a partition satisfying the variant’s condition does not guarantee that every even integer has a partition into two primes.

The number 16 has both kinds of partition, so it already satisfies the original conjecture. For all other even integers greater than 2, however, we still need to prove that we can always find two primes, beyond merely finding two numbers with odd counts of prime factors. That is what the variant’s proof has not established.

How Should We Objectively Evaluate This AI Result?

Evaluating this achievement requires distinguishing between the formal correctness of the proof and the actual scope of the proposition proven:

1. Does the proof of the stated proposition pass verification? Lean checks the proposition explicitly written in the code and the reasoning used to prove it. A third-party developer has independently run and checked this variant’s proof, confirming that no proof steps were skipped.
2. Is the stated proposition the original conjecture? Even if the proof passes verification, we must ask whether it requires “two primes” or the broader category of “two numbers, each with an odd number of prime factors.” This work proves the latter, so it cannot be described as solving the original Goldbach conjecture.

Conclusion: An Odd Number of Prime Factors Is Still Not a Prime

Since 1742, research has advanced along different routes: the circle method proved that every odd integer greater than 5 is the sum of three primes; Chen’s theorem guarantees that every sufficiently large even integer is the sum of a prime and another number with at most two prime factors; and computers have verified that every even integer greater than 2 and no greater than $4 \times 10^{18}$ is the sum of two primes.

The 2026 proof claimed to have been produced by AI allows each summand to have an odd number of prime factors, admitting some composite numbers as well. It proves a proposition with relaxed conditions, but whether every even integer greater than 2 can be written as the sum of two primes remains an open question.