Goldbach's conjecture

The statement. 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 100 = 3 + 97 … Every even integer above 2 is the sum of two primes. Proposed in a letter from Goldbach to Euler in 1742.

In plain words. Primes are the multiplicative atoms of the integers; Goldbach says they also generate the even numbers additively, two at a time. It has been checked up to 4 × 10¹⁸.

What you need first. Stage 10 (primes and the prime factorisation), Stage 12 (the analytic tools that gave the partial results).

The parts. (1) Counting: how many ways can 2n be split as p + q? Heuristically about 2n/(ln 2n)² — plenty, so the conjecture “should” be true. (2) The circle method (Hardy–Littlewood), which turns counting representations into an integral of exponential sums. (3) The weak version — every odd number above 5 is a sum of three primes — which the same method actually proves (Helfgott, 2013). (4) Sieve methods, which get as far as “every large even number is a prime plus a number with at most two prime factors” (Chen, 1973).

What is known. The weak conjecture is a theorem; Chen's theorem stands; the strong conjecture is verified to 4 × 10¹⁸ and unproved.