Goldbachs

Goldbachs refers to conjectures and statements named after the 18th-century Prussian mathematician Christian Goldbach. In mathematics, the most prominent Goldbachs are two conjectures about expressing integers as sums of primes. Together, they form a central topic in additive number theory.

The strong Goldbach conjecture, or simply Goldbach’s conjecture, asserts that every even integer greater than 2 can be written as the sum of two primes. The two primes may be the same (for example, 4 = 2 + 2). Despite extensive numerical verification and partial theoretical progress, the conjecture remains unproven in full generality. It has been checked by computer up to at least 4 × 10^18, but a general proof has not been found.

The weak Goldbach conjecture, also called the ternary Goldbach conjecture, states that every odd integer greater than 5 is the sum of three odd primes. This version was proven by Harald Helfgott in 2013–2014, completing a long line of partial results that began with Vinogradov’s work in the 1930s. Helfgott’s proof established the conjecture as a theorem for all odd integers greater than 5.

Christian Goldbach proposed these ideas in a 1742 correspondence with Leonhard Euler, outlining the idea that every even number could be built from primes and suggesting a broader prime-additive structure of integers. The Goldbachs have since become foundational problems in number theory, illustrating the deep connections between primes and additive representations.

In modern mathematics, the strong conjecture remains open, while the weak conjecture is proven. The topic continues to inspire computational investigations and theoretical advances in prime distribution and additive problems.