Goldbach's conjecture

Goldbach's conjecture is the proposition that every even integer greater than 2 can be expressed as the sum of two prime numbers. Proposed by Christian Goldbach in a 1742 letter to Leonhard Euler, it remains one of the oldest unsolved problems in mathematics — verified computationally for all even numbers up to at least 4 × 10^18 but unproven in general.

The conjecture is of philosophical interest beyond number theory because of its role in discussions of intuitionism and the Law of Excluded Middle. Classically, the statement 'every even integer greater than 2 is the sum of two primes, or some even integer greater than 2 is not the sum of two primes' is trivially true by excluded middle — it is a tautology. Intuitionistically, it is an open problem: neither disjunct has been proved, and therefore the disjunction cannot be asserted. This distinction — between classical tautologies and intuitionistically unresolvable disjunctions — is precisely the gap that Brouwer used to motivate the rejection of excluded middle as a universal logical principle.

The conjecture's durability is itself philosophically interesting. It is not from want of trying: the Hardy-Ramanujan circle method and sieve theory have produced partial results (every even integer is the sum of at most a bounded number of primes; Chen Jingrun proved in 1973 that every sufficiently large even integer is the sum of a prime and a semiprime). But the full conjecture resists proof. Whether this reflects the genuine hardness of the problem or a fundamental limitation of current proof methods — and whether Gödel-type results might make it undecidable in standard arithmetic — remains contested. The possibility that Goldbach's conjecture is true but unprovable in Peano Arithmetic is taken seriously by logicians, though no proof of undecidability is known.