Vinogradov proved in 1937, using the circle method, that every sufficiently large odd number is a sum of three primes — but his proof gave no explicit bound on 'sufficiently large'. Decades of work made the threshold effective and then smaller: by 2002, Liu and Wang had brought it down to about , a number so large that checking every smaller odd number by computer is physically impossible (more than the number of picoseconds since the Big Bang). Helfgott's goal was not to prove a new qualitative statement — that was settled in 1937 — but to shrink the threshold enough that direct computation could take over below it, finally making the full statement unconditionally true for every single odd .
Helfgott's own papers (arXiv:1305.2897 for major arcs, arXiv:1205.5252 and 1305.3062 for minor arcs and the numerical companion) restate the goal precisely: reduce Vinogradov's ineffective 'sufficiently large' to an explicit small enough that a computer verification of all remaining odd becomes feasible within available computing resources — a threshold on the order of –, not . The next three steps carry out the analytic half of this program (the circle method with fully explicit constants); the final step supplies the matching computational half.
- Ineffective vs. effective bound
- An 'ineffective' proof shows a threshold exists but gives no algorithm to compute it; an 'effective' one gives an explicit numerical value, even if impractically large, that can in principle be checked by finite computation.