Unsolved: Goldbach conjecture

The Goldbach conjecture is one of the oldest and easiest-to-understand math problems that remains unsolved. The problem was originally posed to Leonhard Euler in a letter from amateur mathematician Christian Goldbach in 1742. The original form of the conjecture, now known as the weak Goldbach conjecture says:

Every whole number greater than five is the sum of three prime numbers.

Euler restated the problem in an equivalent form, the strong Goldbach conjecture (or just the Goldbach conjecture):

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

So,

4 = 2 + 2
6 = 3 + 3
8 = 3 + 5
10 = 5 + 5
12 = 5 + 7
...
100 = 47 + 53
...

Progress

In 1966, Chinese mathematician Chén Jǐngrùn showed that every sufficiently large even integer is the sum of a prime and semiprime (a number that has at most two prime factors).

Brute force methods have been used to show that the Goldbach conjecture is true for even integers up to about 1.5 * 10¹⁸* (1,500,000,000,000,000,000 or 1.5 billion billion).


Notes:

* Brute force results are as of July 24, 2009. See Goldbach conjecture verification for up-to-date results.