Goldbach’s Conjecture: Proposed in 1742, Goldbach’s Conjecture states that every even integer greater than 2 can be expressed as the sum of two prime numbers (e.g., 10 = 3 + 7). A weaker version—every odd number greater than 5 is the sum of three primes—was proved by Vinogradov for sufficiently large numbers and fully by Helfgott in 2013. However, the strong (binary) conjecture remains unproven. It has been verified computationally up to 4 × 10¹⁸, with no exceptions. The problem is central to additive number theory and relates to the distribution of primes. Techniques like the circle method and sieve theory have yielded partial results (e.g., Chen’s theorem: every large even number is a sum of a prime and a semiprime), but a complete proof eludes. Goldbach’s Conjecture exemplifies how prime numbers, though fundamental, still guard deep secrets about their additive structure.
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