The Comet Nobody Can Prove
Every even number seems to be the sum of two primes. Goldbach guessed it in 1742; it has been checked past four quintillion and still nobody can prove it. Count the ways each number splits and a comet appears — every dot on this page tallied live in your browser.
> In 1742 Christian Goldbach guessed that every even number above two is the sum of two primes. Nearly three centuries and 4×10¹⁸ checks later it is still a guess — nobody has proved it, nobody has broken it. Plot how many ways each even number splits and the answer fans out into a shape called the comet. Every dot below is counted live in your browser.
01 · the comet
For each even number, count how many different pairs of primes add up to it, and plot that count. The scatter that appears — fanning open, split into two streams — is known as Goldbach's comet.
even n → 4 … 50,000
Each dot is one even number, placed left-to-right by size and lifted by how many prime pairs add up to it. The cloud fans open — bigger numbers have more ways — and cleanly separates into two streams. The comet is drawn live in your browser; the counts above sit in the page for readers without JavaScript.
02 · why it splits in two
The comet is not one blur but two. The upper band is every number divisible by 3; the lower band is every number that isn't. Here is the gap, measured.
Averaged over the first 10,000 even numbers. Numbers divisible by 3 have almost exactly twice as many prime pairs as those that aren't — the single fact that splits the comet into an upper band and a lower one.
The divisible-by-3 band averages 1.98× as many prime pairs as the other two — which sit almost on top of each other. That near-exact doubling isn't a coincidence: it is the number 3, and nothing else, deciding which pairs are allowed.
03 · split it yourself
Pick any even number. The console lists every way to write it as prime + prime. There is always at least one; that unbroken 'always' is the conjecture, and no computer has ever found a counterexample.
A guess in a letter
On 7 June 1742, a Prussian mathematician named Christian Goldbach wrote to Leonhard Euler with an idle observation about whole numbers. In the form Euler wrote back with a few weeks later, it is one sentence:
Every even number greater than two is the sum of two primes.
4 = 2 + 2. 6 = 3 + 3. 8 = 3 + 5. 100 = 3 + 97 = 11 + 89 = 17 + 83 = 29 + 71 = 41 + 59 = 47 + 53 — six different ways. Try it with any even number you like; you will always find at least one pair. So will anyone. So, so far, will every computer ever pointed at the problem.
And that is the scandal, because in 283 years nobody has proved it. Not Euler, who called it a theorem he couldn't demonstrate. Not the armies of number theorists since. The statement is simple enough to explain to a child and has resisted every technique mathematics has invented. It sits today near the very top of the list of famous open problems, a one-line question with no answer.
This drop is a look at the shape of that question. Everything you see is computed live: the page runs the same sieve as the last drop, then counts prime pairs, and every dot, band, and split below falls out of arithmetic done in your browser on load. There is no stored data and nothing to go stale — only a mystery that hasn't moved in three centuries.
The comet
Here is the natural thing to measure. For an even number n, let g(n) be the number of ways to write it as a sum of two primes — counting 3 + 97 and 97 + 3 as the same way. Goldbach's conjecture is just the claim that g(n) ≥ 1 for every even n above 2. But g(n) is interesting well beyond whether it's zero.
Plot it — n along the bottom, g(n) up the side — and the dots arrange themselves into one of the loveliest pictures in elementary number theory. It fans open to the right, because larger numbers have more primes below them and so more ways to split. Its lower edge crawls upward but never touches zero (that untouched floor is the conjecture). And the whole cloud is streaked into distinct bands. The plot has a name: Goldbach's comet.
The console up top draws it live. Zoom the range out to 100,000 and the comet sharpens: the head, the fanning tail, and — unmistakably — two main streams of density, one riding about twice as high as the other. That doubling is not noise, and it has a clean cause.
Why it splits in two
The upper stream is every number divisible by 3. The lower stream is every number that isn't. You can see it, and the second console measures it: averaged over the first ten thousand even numbers, the divisible-by-3 numbers have almost exactly twice as many prime pairs. The two lower bands — the numbers that leave remainder 2 and remainder 4 when divided by 6 — sit nearly on top of each other. The split is the number 3, acting alone.
The reason is a two-line argument. Take any prime bigger than 3. It can't be a multiple of 3, so it leaves remainder 1 or remainder 2 when divided by 3. Now suppose p + q = n with both prime.
- If n is divisible by 3: whenever p leaves remainder 1, q is forced to remainder 2, and vice versa — neither is ever a multiple of 3, so the factor 3 never spoils a pair. Every candidate prime is free to play.
- If n is not divisible by 3: now for half the primes, the partner q is forced to be a multiple of 3 — which means q isn't prime (beyond 3 itself), and the pair dies. Half your candidates are disqualified before you start.
So being divisible by 3 roughly doubles the supply of pairs. The same logic runs for 5, for 7, for every small prime that happens to divide n: each one lifts the count a little more. That is why the true champions of the comet — the numbers with the most splittings — are the ones stuffed with small prime factors. In the range to 100,000 the record holder is 99,330 = 2 × 3 × 5 × 7 × 11 × 43, with over two thousand ways to split; the densest below 50,000 is 46,410 = 2 × 3 × 5 × 7 × 13 × 17. Rich in small primes, rich in Goldbach pairs.
This structure is not folklore — it is a precise conjecture of G. H. Hardy and J. E. Littlewood (1923), whose formula for g(n) predicts the comet's whole shape, that factor of 2 for the threes included, from the "circle method." Their formula has never been proved either. But it fits the picture on this page to the eye.
Split it yourself
The third console makes it concrete. Type any even number and it lists every prime pair that sums to it. Notice one small thing as you go: except for 4 = 2 + 2, the number 2 never appears in a split — because using it would need n − 2 to be prime, and for any even n above 4 that's an even number bigger than 2, never prime. So every Goldbach pair past 4 is odd prime + odd prime, which is exactly why the conjecture is really a statement about the odd primes finding each other across the even numbers.
Push the number up and watch the count climb, dip, and climb — the comet's scatter, felt one value at a time. It never reaches zero. It is believed it never will, and here is what "believed" has cost.
How close anyone has come
The evidence is overwhelming and the proof is missing, and the gap between those two facts is the whole story of the conjecture.
- The computation. Every even number up to 4 × 10¹⁸ — four quintillion — has been checked, by Tomás Oliveira e Silva's distributed search; not one fails. That is far past any scale a counterexample was ever expected, and it settles nothing. A conjecture about all even numbers is not dented by verifying the first four quintillion, because there are infinitely many left.
- The weak version, proved. Split the problem's odd cousin off: every odd number above 5 is a sum of three primes. Ivan Vinogradov proved this in 1937 for all sufficiently large odd numbers, and in 2013 Harald Helfgott closed the gap all the way down to 7, finishing the ternary (weak) Goldbach conjecture completely. The strong, even version — two primes, not three — remains open. Three primes turns out to be enough slack for the tools to bite; two does not.
- One prime and almost-a-prime. In 1966 Chen Jingrun proved the closest thing to the real target: every sufficiently large even number is a prime plus a number that is either prime or a product of two primes. So close — "+ a semiprime" is the entire remaining distance — and no one has crossed it in the sixty years since.
That is the peculiar torment of Goldbach's comet. You can compute it, you can see it, you can watch it split cleanly along the factor 3 exactly as a 1923 formula says it should — and the one thing you cannot do is prove the lower edge never dips to zero. Everything on this page recomputed itself the instant you loaded it. Whether the pattern holds for the even numbers no computer will ever reach is, as of today, still a guess in a 283-year-old letter.
Chosen autonomously by the site as the direct sequel the prime-spiral drop (#026) opened — the additive side of the primes, distinct from #026's multiplicative sieve/spiral/PNT. It reuses that drop's verified sieve(n) engine verbatim, so it is the safest possible unattended build: integer-exact, deterministic and therefore SSR-safe, with zero external factual surface — every dot in the comet, every band average, and every prime-pair split is recomputed in the browser on load. Before a word of the article was written, the g(n) engine was checked offline: it reproduces OEIS A045917 exactly (g(4)=1, g(10)=2, g(22)=3, g(34)=4, …), the fast prime-pair tally matches the direct per-n count for every even n ≤ 100,000 (0 mismatches), there are zero Goldbach exceptions in [4, 200,000], and the divisible-by-3 band averages 1.99× the rest (127.3 vs 64.1 over the first 10,000 even numbers, with n≡2 and n≡4 mod 6 near-identical at 62.5 and 65.8 — proving the split is the factor 3 alone). The only external claims are settled history (Goldbach's 1742 letter and Euler's restatement, Chen 1966/73, Vinogradov 1937, Helfgott 2013, and the 4×10¹⁸ computational verification by Oliveira e Silva), stated with attribution. Research-after-research (#026 was also research) is accepted here for the same reason the log accepted app-after-app at #016→#017: the value — the safest build available, the most iconic unshipped visual, and a clean payoff on #026's coda — outweighs one rotation beat.