On 7 June 1742, Christian Goldbach, a Prussian mathematician who had tutored the young Tsar Peter II, wrote a letter to Leonhard Euler and tossed off a guess in the margin: every whole number greater than 2, he suggested, could be written as a sum of primes. Euler wrote back sharpening it into the form we know today — every even number greater than 2 is the sum of two primes — and added that he regarded it as “a completely certain theorem”, although he could not prove it. Neither could anyone else. Nearly three centuries later, the greatest mathematician of his age still owes us that proof, and Goldbach’s conjecture sits exactly where it started: obviously true, and unproven.
Try it yourself, because that is the seduction. Four is 2 + 2. Six is 3 + 3. Eight is 3 + 5. Ten is 3 + 7, or 5 + 5. One hundred is 3 + 97, or 11 + 89, or 17 + 83. Computers have marched this check up to 4 × 10¹⁸ — four billion billion — without finding a single exception. Worse, the bigger the even number, the more ways there seem to be to split it, which is precisely why mathematicians believe the conjecture and precisely why belief is not the job. A proof must cover every even number to infinity, and primes are stubborn creatures: they are defined by multiplication, and addition is a language they were never built to speak.
The near misses are heroic. In 1937 the Soviet mathematician Ivan Vinogradov proved that every sufficiently large odd number is the sum of three primes, and in 2013 the Peruvian mathematician Harald Helfgott completed the so-called weak Goldbach conjecture for all odd numbers above five. The strong version has resisted everything. The closest approach remains Chen Jingrun’s 1973 theorem that every sufficiently large even number is the sum of a prime and a number with at most two prime factors — a result Chen produced in China during the Cultural Revolution, working in a tiny room under persecution, which later made him a national hero with a postage stamp to his name. In 2000, to publicise the novel Uncle Petros and Goldbach’s Conjecture, the publisher Faber offered a million dollars for a proof within two years. Nobody collected.
The problem you can teach in ten seconds
The other great child-friendly monster is younger and stranger. In 1937 the German mathematician Lothar Collatz began playing with a rule you can teach to a seven-year-old. Take any positive whole number. If it is even, halve it. If it is odd, triple it and add one. Repeat. Start with 7 and you get 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. The Collatz conjecture says that wherever you start, you always crash down to 1 in the end. The sequences are nicknamed hailstone numbers, because they rise and fall like hailstones in a storm cloud before finally hitting the ground — the modest starting value 27 soars to a peak of 9,232 and takes 111 steps to land.
The problem has collected aliases the way folklore does — the 3n + 1 problem, the Syracuse problem, Ulam’s conjecture, Kakutani’s problem — because it swept through mathematics departments in the 1950s and 60s like a rumour. Shizuo Kakutani joked that it was a conspiracy to slow down American research, since everyone dropped what they were doing to try it. Paul Erdős delivered the most quoted verdict in the field: “Mathematics is not yet ripe for such problems”, and put up 500 dollars of his own money for a solution. Computers have verified the conjecture for every starting number up to beyond 2⁶⁸ — roughly 300 billion billion — and John Conway proved that a generalised version of the game is formally undecidable. The deepest modern progress came in 2019, when Terence Tao proved that almost all starting numbers eventually fall almost as low as you like: in his own summary, about as close as one can get to the Collatz conjecture without actually solving it.
Why are these two so hard? Because both mix arithmetic’s two personalities. Goldbach asks an additive question about multiplicative objects. Collatz alternates halving with tripling, and the parity of what comes next behaves like a coin toss; the sequence is deterministic, yet it wears the costume of pure randomness. Nothing in the mathematician’s toolkit gets a firm grip on either, which is why both problems function as a famous warning to the young and ambitious — and as a lure. Every few years a claimed proof of Goldbach or Collatz surfaces, and every few years it quietly dissolves.
There is something almost comforting in that. The frontier of mathematics is not hidden behind graduate textbooks; it can be written on a beer mat, checked by a child, and defended by nothing except its own bottomless depth. Anyone can play. No one, so far, can win.
Quiz nuggets
- Goldbach’s conjecture began in a letter from Christian Goldbach to Leonhard Euler dated 7 June 1742; the modern form says every even number above 2 is a sum of two primes.
- The conjecture has been computer-verified up to 4 × 10¹⁸; the weak (odd-numbers) version was proved by Harald Helfgott in 2013.
- Chen Jingrun proved in 1973 that every sufficiently large even number is a prime plus a number with at most two prime factors.
- The Collatz conjecture (halve if even, triple and add one if odd) is named after Lothar Collatz, who studied it in 1937; starting from 27 takes 111 steps and peaks at 9,232.
- Paul Erdős said “mathematics is not yet ripe for such problems” and offered a 500-dollar prize; publisher Faber offered 1 million dollars for Goldbach in 2000, unclaimed.