Goldbach's Conjecture

On 7 June 1742 the amateur mathematician Christian Goldbach wrote a letter to Leonhard Euler with an idle-looking observation about the whole numbers. Euler polished it into the crisp statement we still stare at today:

\text{every even number } N \ge 4 \text{ is a sum of two primes.}

Check it and it never fails: 4=2+2, 6=3+3, 8=3+5, 10=3+7=5+5, 100=3+97=47+53. A modern computer has confirmed it for every even number up to 4\times10^{18} without a single exception. And yet — nearly three centuries on — nobody has proved it. It is one of the oldest and most famous open problems in all of mathematics: a sentence a ten-year-old understands, guarding a secret no analytic number theorist has cracked.

This page tells the story of what we do know — because the partial results are spectacular, and one close cousin of Goldbach's guess has, since 2013, been a fully proved theorem.

Two conjectures, not one

Goldbach's idea splits into a strong claim about even numbers and a weak claim about odd numbers. It is vital to keep them apart, because their fates could not be more different.

Every even integer N \ge 4 can be written as N = p + q with p, q prime.
Status: unproved, and widely believed to be out of reach of current methods.

Every odd integer N \ge 7 can be written as N = p_1 + p_2 + p_3, a sum of three primes.
Status: a theorem — Vinogradov (1937) for all large N, completed for every N \ge 7 by Harald Helfgott in 2013.

The names give away the relationship: it is called "weak" because the strong conjecture implies it. If every even number is a sum of two primes, then any odd N \ge 7 can be handled by peeling off a 3 and writing N = 3 + (N-3), where N-3 is even and \ge 4, hence itself two primes. So proving the strong conjecture would give the weak one for free. The reverse does not hold — and that asymmetry is the heart of this whole subject.

A wrinkle of history

Goldbach's original 1742 wording was slightly different — and, by today's rules, slightly wrong. He counted 1 as a prime and conjectured that every integer greater than 2 is a sum of three primes. Euler replied that this was equivalent to the cleaner statement that every even number is a sum of two primes, which he called "a theorem I regard as entirely certain, although I cannot prove it." Nearly 300 years later, Euler's confidence still stands unvindicated by proof. Modern convention drops 1 from the primes, which is why we quote the ranges N \ge 4 (binary) and N \ge 7 (ternary).

Worked example: counting the representations of 100

Instead of just asking whether an even number is a sum of two primes, let's count how many ways. Write r(N) for the number of unordered prime pairs \{p, q\} with p \le q and p + q = N. To find r(100), walk the primes p up to 50 and test whether 100 - p is also prime:

p	100 - p	both prime?
3	97	yes ✓
11	89	yes ✓
17	83	yes ✓
29	71	yes ✓
41	59	yes ✓
47	53	yes ✓

(The primes 5, 7, 13, 19, 23, 31, 37, 43 all fail — e.g. 100-7=93=3\times31.) So 100 has exactly six Goldbach representations: r(100) = 6. Goldbach's conjecture only asks that r(N) \ge 1 for every even N \ge 4 — but the data suggest something far stronger: as N grows, r(N) doesn't just stay positive, it grows, with room to spare.

How many representations should there be? The Hardy–Littlewood heuristic

In 1923, G. H. Hardy and J. E. Littlewood turned the counting into a prediction using the circle method. The idea is a probabilistic one. A number near N is prime with "chance" about 1/\ln N (the Prime Number Theorem), so if primality were independent, a pair p, N-p would both be prime with chance about 1/(\ln N)^2, giving roughly N/(\ln N)^2 representations. Primality is not quite independent — divisibility by small primes correlates the two — and correcting for that introduces a singular series factor:

The number of Goldbach representations of an even N is expected to satisfy

r(N) \sim 2\,\Pi_2 \prod_{\substack{p \mid N \\ p > 2}} \frac{p-1}{p-2}\;\cdot\; \frac{N}{(\ln N)^2},

where \Pi_2 = \displaystyle\prod_{p>2}\left(1 - \frac{1}{(p-1)^2}\right) \approx 0.6601618 is the twin-prime constant.

Worked check at N = 100. Since 100 = 2^2 \cdot 5^2, the only odd prime dividing it is 5, contributing the factor \tfrac{p-1}{p-2} = \tfrac{5-1}{5-2} = \tfrac43. With \ln 100 \approx 4.605, so (\ln 100)^2 \approx 21.21:

r(100) \approx 2(0.6602)\left(\tfrac43\right)\frac{100}{21.21} \approx 8.3.

The heuristic predicts about 8; we counted 6. That is strikingly good agreement for a probabilistic guess at a number as small as 100, where the leading term N/(\ln N)^2 is only a crude stand-in for the sharper logarithmic integral \int_2^N \frac{dt}{\ln t\,\ln(N-t)}. The bigger N gets, the better the formula fits — and, crucially, it drives r(N) off to infinity, so the conjecture isn't just true, it's true with an enormous and growing margin.

Notice the extra factor for numbers divisible by small primes: an even N that is a multiple of 3 gets the boost \tfrac{3-1}{3-2} = 2, so multiples of 6 have roughly twice as many representations as their neighbours. That single fact is the secret behind the shape of the picture we plot next.

The Goldbach comet

Plot r(N) against every even N and a gorgeous structure appears — a spray of points fanning upward that number theorists call the Goldbach comet. Every single even number from 4 onward sits at height \ge 1 (that is the conjecture, holding for every point you can see), and the whole cloud drifts steadily higher as N grows.

Look closely and the comet is striped: it separates into ribbons. The top ribbon is the multiples of 6 — exactly the doubling we predicted from the singular series — while even numbers not divisible by 3 trail along a lower band. The heuristic doesn't just estimate the average height of the comet; it explains its internal anatomy.

Why the strong conjecture is out of reach

The circle method that solved the ternary problem hits a wall on the binary one. The method writes the number of representations as an integral over the circle [0,1), split into major arcs (near rationals with small denominators, where the sum is large and computable) and minor arcs (everywhere else, which must be shown to contribute only a small error).

Three variables give you room; two do not. For the ternary problem (Vinogradov's three-primes theorem) there are three prime variables, and the minor-arc integral can be bounded — one factor tames the others. For the binary problem there are only two, and the minor arcs cannot be controlled: the error term is as large as the main term. The circle method simply doesn't converge for two variables.
The parity problem bites. Even sieve methods run into the notorious parity problem: classical sieves cannot tell numbers with an even number of prime factors from those with an odd number, so they can never, on their own, force a summand to be exactly prime (one prime factor). This is precisely why Goldbach resists being finished off by sieving.

The consensus is that a proof of binary Goldbach would need a genuinely new idea — not a sharpening of the circle method, but something that breaks the parity barrier. That idea does not yet exist.

The great partial results

If we can't reach "a sum of two primes," we can get astonishingly close. Each of the following is a hard-won theorem.

Ternary Goldbach is done. Vinogradov (1937) proved every sufficiently large odd number is a sum of three primes; Helfgott (2013) closed the remaining finite gap, so it holds for all odd N \ge 7.
Chen's theorem (1973). Every sufficiently large even N is p + P_2: a prime plus a number that is either prime or a product of two primes (a "semiprime"). This is the closest anyone has come to the real thing.
Ramaré (1995). Every even number is a sum of at most 6 primes. (Tao later showed every odd number is a sum of at most 5 primes, now subsumed by Helfgott's result.)
Numerical verification. Binary Goldbach has been checked directly for every even N up to 4\times10^{18}.

Line them up and the gap is agonisingly thin. Chen gives you "prime plus almost-prime"; Goldbach wants "prime plus prime." Between P_2 and a single prime lies the entire unsolved problem — and the parity barrier standing in the way.

This is the single most common confusion in the whole topic, so pin it down:

The weak / ternary conjecture (odd N = three primes) is a proved theorem since Helfgott (2013).
The strong / binary conjecture (even N = two primes) remains open. Proving the weak one did not settle it.

And be just as careful with the computer evidence: verifying every even number up to 4\times10^{18} is not a proof. There are infinitely many even numbers, and a conjecture can hold for the first quintillion cases and fail at the next — the history of number theory has examples where the smallest counterexample is astronomically large. A finite check, however heroic, can only ever disprove a conjecture (by finding a counterexample), never prove one.

Because the Hardy–Littlewood heuristic predicts r(N) \sim 2\Pi_2 \cdots N/(\ln N)^2, which races off to infinity. For Goldbach to fail, some even number would need r(N) = 0 — but the expected count near, say, N = 10^{18} is astronomically large (billions of representations). For all of those independent-looking prime pairs to conspire and vanish simultaneously would be a coincidence of unimaginable improbability. That's not a proof — coincidences of exactly this kind are what the parity problem warns us we can't rule out — but it is why essentially every number theorist would bet their house that Goldbach is true.