Goldbach's conjecture

Goldbach's conjecture states that every even integer greater than 2 is the sum of two prime numbers. It remains unproved despite extensive computational verification.

Prime number: A natural number greater than 1 whose only positive divisors are 1 and itself. The conjecture asks whether every eligible even number can be built from two primes.
Goldbach partition function: The function that counts unordered representations of an even integer as a sum of two primes. A positive value for every even integer above 2 would establish the conjecture.
Christian Goldbach: An eighteenth-century Prussian mathematician known for his correspondence with Leonhard Euler about prime representations. His 1742 letter to Euler contained the claim that became the conjecture.
Chen's theorem: A theorem stating that every sufficiently large even integer is the sum of a prime and a number with at most two prime factors. It approaches Goldbach's target while allowing the second summand to be almost prime.
Goldbach's conjecture computational verification: Algorithmic testing of Goldbach representations for finite ranges of even integers. Verification has reached vast ranges, yet every finite check leaves infinitely many cases.
Even number: An integer divisible by 2, equivalently an integer of the form 2k for some integer k. Evenness is the precise condition on the integers covered by the conjecture.
Sieve theory: A collection of methods that estimate or bound integers satisfying divisibility restrictions. Sieve methods constrain candidate prime pairs but have not settled the binary conjecture.
Leonhard Euler: An eighteenth-century Swiss mathematician whose work shaped many areas of modern mathematics. Euler reformulated Goldbach's claim into the familiar assertion about two primes summing to an even number.
Ramaré's theorem: A theorem proving that every even integer is a sum of at most six primes. It proves a bounded-summand result, weaker than the conjecture's two-prime claim.
Twin prime conjecture: The conjecture that infinitely many prime pairs differ by 2. It is another famous unresolved statement about the distribution of prime numbers.

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