Worked solution: Closing the gap: the ternary Goldbach conjecture, unconditionally (Helfgott, 2013)

Step 1 of 5: From an astronomically large threshold to a computable one

In plain words

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 $C\approx e^{3100}\approx 10^{1342}$ , 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 $C$ enough that direct computation could take over below it, finally making the full statement unconditionally true for every single odd $n>5$ .

\forall\, n>5 \text{ odd}: \quad n = p_1+p_2+p_3, \qquad \text{Vinogradov (1937): true for } n > C \ (\text{ineffective, later } C\approx 10^{1300})

Detailed analysis

Vinogradov's 1937 proof showed that the weighted representation count $R(n)$ of the next step is positive for all sufficiently large odd $n$ , but 'sufficiently large' could not be computed from the argument: the major-arc estimates have to allow for a possible exceptional real zero of a Dirichlet $L$ -function $L(s,\chi)$ , and Siegel's theorem keeps such a zero away from $s = 1$ only by an ineffective constant. Making the threshold effective and then small took decades: Borozdkin first gave a computable but astronomical bound, and the sharpest pre-Helfgott result, due to Liu and Wang (2002), required $n > e^{3100} \approx 10^{1346}$ — the $10^{1300}$ -scale constant quoted in the display.

A threshold of that size is permanently beyond enumeration, so Helfgott's program (preprints arXiv:1205.5252, 1305.2897 and 1305.3062, consolidated in arXiv:1312.7748) was quantitative from the outset: redo the entire circle method with every implied constant replaced by an explicit, moderately sized one, so that the cutoff between 'proved analytically' and 'checked numerically' lands near $10^{27}$ – $10^{30}$ instead of $10^{1346}$ . Three ingredients make the gain possible, one per following step: fully explicit major-arc asymptotics driven by computed data on the zeros of $L(s,\chi)$ ; minor-arc bounds on $S(\alpha)$ that do not lose extra logarithmic factors; and a numerical verification engineered to overshoot the analytic threshold.

The proof therefore has two halves built to overlap: the analytic machinery must cover every odd $n \ge 10^{27}$ , and a separate computation must cover $5 < n \le 10^{27}$ with a wide margin. That the two halves do in fact meet — with room to spare — is the content of the last step.

Terms in this step

Ineffective vs. effective bound: An 'ineffective' proof shows a threshold $C$ 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.