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Goldbach's conjecture

Every even number greater than 2 is the sum of two primes.

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.

Symbols used here

\blacksquare\ \text{or}\ \square
end of proof (halmos)
Marks the point where the statement has been established.
\varphi(n),\ \pi(x)
Euler's totient, prime-counting function
Count of 1..n coprime to n; number of primes up to x.
\zeta(s)
Riemann zeta function
Σ 1/nˢ, continued to the complex plane; its zeros are the Riemann Hypothesis.
O(n^2),\ \Theta,\ \Omega
big-O notation
Grows no faster than n² (up to a constant), for large n.
\frac{\partial u}{\partial t},\ \nabla^2 u
partial derivative in time, Laplacian
Rate of change in time; sum of second partials (the diffusion operator).

Questions people ask

Can I actually work on these?

You can understand them, which is the honest first step and what these pages are for. Working on them means the full path through the stages above and then the research literature — but every person who has made progress started by reading the statement.

Why are they unsolved if so many people have tried?

Usually because the existing tools provably cannot work (the barriers in P vs NP), or because the problem mixes two structures mathematics handles separately (additive and multiplicative in Goldbach and Collatz). A solution needs a genuinely new idea.

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Vairāk Frontiers

The Riemann HypothesisNavier–Stokes existence and smoothnessP versus NPThe Collatz conjectureThe twin prime conjecture