Strong Goldbach Conjecture

Determine whether every even integer greater than two can be expressed as the sum of two prime numbers by either proving the conjecture in full generality or producing a counterexample.

Background

The paper uses the strong Goldbach Conjecture as a central case paper to illustrate Gowers’ no-coincidence and no-miracles principles for how mathematicians form justified beliefs about unproved statements. The author notes extensive computational verification of the conjecture up to very large bounds and discusses statistical heuristics about the distribution of primes that support the conjecture’s plausibility.

This example anchors the broader argument about the relationship between "good reasons" and efficiently findable proofs in mathematics, contrasting the empirical evidence and heuristic justification for the conjecture with the proof-complexity perspective that many true statements may lack reasonably short proofs.