Goldbach's Conjecture: The Additive–Multiplicative Bridge

In this Deep Dive we unpack the deceptively simple claim that every even number greater than 2 is the sum of two primes. We explore why the problem sits at the crossroads of addition and multiplication, how tools like the circle method and sieve theory struggle with the parity barrier, and what that means for a full proof. Along the way we trace key milestones—from Helfgott’s work on the weak Goldbach conjecture to Chen Jingrun’s prime‑plus‑almost‑prime breakthroughs and near‑misses that replace two primes with a prime plus powers of two—before outlining what remains to close the gap and why the mystery endures.

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