Goldbach's Conjecture

Statement

Goldbach (1742): every even integer n>2 can be expressed as the sum of two primes. Verified up to 4×10¹⁸; unproven in general.

Examples

In Depth

Goldbach's conjecture (1742): every even integer greater than 2 is the sum of two primes. Verified computationally up to \(4\times10^{18}\), it remains unproved. It is one of the oldest and most famous unsolved problems in mathematics.

The weak Goldbach conjecture (every odd integer \(>5\) is the sum of three primes) was proved by Helfgott in 2013 using the Hardy–Littlewood circle method with extensive numerical verification for small cases.

Vinogradov's theorem (1937): every sufficiently large odd integer is the sum of three primes. 'Sufficiently large' originally meant \(>e^{e^{11.503}}\), a number with about \(10^{6846}\) digits. Helfgott's work reduced this to \(>5\), completing the proof for all odd integers \(>5\).

The Hardy–Littlewood conjecture gives an asymptotic for the number of representations of \(2n\) as a sum of two primes: \(r(2n)\sim 2C_2\prod_{p\mid n,p>2}\frac{p-1}{p-2}\cdot\frac{2n}{(\ln 2n)^2}\). This is consistent with numerical data but far from proved.

Chen's theorem (1966): every sufficiently large even integer is the sum of a prime and a number with at most two prime factors (a 'P2 number'). This is the closest proved result to Goldbach's conjecture and uses the large sieve and Brun's sieve.

Key Properties & Applications

Numerical verification: Goldbach's conjecture has been verified for all even integers up to \(4\times10^{18}\) by Oliveira e Silva et al. (2013). The verification uses a segmented sieve and took years of distributed computation.

The Goldbach comet: plotting the number of ways to write \(2n\) as a sum of two primes shows a comet-like pattern. The 'tail' corresponds to even numbers with few representations; the 'head' to those with many. The Hardy–Littlewood conjecture predicts the average number of representations.

Schnirelmann's theorem (1930): every integer \(\geq2\) is a sum of at most \(C\) primes for some absolute constant \(C\). This was the first unconditional result toward Goldbach. The current best unconditional result (Helfgott, 2013) gives \(C=3\) for odd integers \(>5\).

Further Reading & Context

Goldbach's original letter to Euler (1742) stated a slightly different conjecture: every integer \(>2\) is the sum of three primes (counting 1 as prime, as was common then). The modern 'strong' Goldbach conjecture (every even \(>2\) is a sum of two primes) is due to Euler's reformulation.

The ternary Goldbach conjecture (every odd \(>5\) is a sum of three primes) was proved by Helfgott in 2013. The proof combines the Hardy–Littlewood circle method with extensive numerical verification for small cases (up to \(8.875\times10^{30}\)) and careful analytic estimates for larger values.

Schnirelmann density and additive bases: a set \(A\) of non-negative integers is an additive basis of order \(h\) if every sufficiently large integer is a sum of \(h\) elements of \(A\). Schnirelmann proved the primes form an additive basis; the Goldbach conjecture would imply order 2 for even integers.

The Goldbach conjecture is related to the distribution of prime pairs. The Hardy–Littlewood conjecture gives an asymptotic for the number of representations \(r(2n)\) of \(2n\) as a sum of two primes. Numerical evidence strongly supports both the conjecture and the asymptotic formula.

Computational verification uses a segmented sieve: for each even \(2n\leq N\), check if \(2n-p\) is prime for each prime \(p\leq n\). The verification up to \(4\times10^{18}\) required petabytes of computation and confirmed the conjecture for all tested values.