Goldbach's conjecture
Goldbach's conjecture, also called the strong or binary Goldbach conjecture, states that every even integer greater than 2 can be written as the sum of two prime numbers. Despite its simple statement and extensive computational checks, no general proof is known. It is a central open problem in additive number theory, distinct from the weak (ternary) Goldbach conjecture, which has been proved.
Statement and examples
The conjecture states that every even integer greater than 2 is the sum of two primes. A representation of an even number as a sum of two primes is called a Goldbach partition. For example, 10 = 3 + 7 = 5 + 5, and 28 = 5 + 23 = 11 + 17. The claim includes 4 = 2 + 2; for any even integer greater than 4, both summands must be odd primes. Some numbers have many partitions, and the conjecture does not prescribe a unique one.
History
The conjecture is associated with an exchange between Christian Goldbach and Leonhard Euler in 1742. Goldbach's original statement used the then-common convention that 1 counted as a prime and concerned writing integers greater than 2 as sums of three primes. Euler reformulated the problem in the equivalent modern form for even integers as sums of two primes, using today's definition that excludes 1 from the primes. The problem has since become a well-known question in number theory.
What is known
Extensive computation has verified the conjecture for every even integer up to 4 × 10¹⁸, but checking a finite range cannot establish the claim for all even integers. Important partial results fall short of a proof of the two-prime statement. In 1973, Chen Jingrun proved that every sufficiently large even number is the sum of a prime and a number with at most two prime factors. The related weak, or ternary, Goldbach conjecture says that every odd integer greater than 5 is a sum of three primes; Harald Helfgott proved this result. The weak theorem does not prove the strong conjecture.
Mathematical significance
Goldbach's conjecture is a problem in additive number theory, which studies how numbers can be represented as sums of elements from specified sets. Its simple formulation contrasts with the difficulty of proving that primes supply a suitable pair for every even number. Computational searches and analytic methods have advanced understanding of prime representations, but neither examples over a finite range nor results about almost all even numbers alone resolve every case. The distinction between an experimentally verified range, partial theorems, and a proof for all integers is essential when describing its status.
Key facts
- Statement
- Every even integer greater than 2 is conjectured to be a sum of two primes.
- Other name
- The two-prime version is also called the strong or binary Goldbach conjecture.
- Status
- It remains an open problem; no general proof is known.
- History
- The conjecture is associated with a 1742 exchange between Christian Goldbach and Leonhard Euler.
- Computational check
- Published computational work reports verification for even integers up to 4 × 10¹⁸; this is not a proof for all integers.
- Related theorem
- Helfgott proved the weak (ternary) Goldbach conjecture for odd integers greater than 5; it is distinct from the strong version.
- Partial result
- Chen's theorem shows that every sufficiently large even integer is a prime plus a number with at most two prime factors.
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References
- 1.Goldbach conjecture
Encyclopaedia Britannica · other · reliability: unverified · retrieved 30 October 2026
Supports: Defines the modern two-prime assertion and summarizes its 1742 origin and partial results, including Chen's theorem.
- 2.Goldbach Conjecture
Wolfram MathWorld · other · reliability: unverified · retrieved 30 October 2026
Supports: Describes the strong or binary statement as open, discusses its relation to Goldbach's original formulation, and distinguishes the proved weak conjecture.
- 3.The ternary Goldbach conjecture is true
arXiv (Harald A. Helfgott, 2013; revised 2014) · other · reliability: unverified · retrieved 30 October 2026
Supports: The author's paper proves the ternary Goldbach conjecture: every odd integer greater than 5 is the sum of three primes.
- 4.Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4⋅10¹⁸
American Mathematical Society, Mathematics of Computation 83 (2014), 2033–2060 · other · reliability: unverified · retrieved 30 October 2026
Supports: Reports the computational verification of the even Goldbach conjecture up to 4 × 10¹⁸; this finite check is not a proof for every even integer.
References are suggested by the research engine and may be incomplete. Unverified URLs are omitted rather than guessed.
Article history
- Version 130 October 2026 · EditorCreated by Zapia