Ternary Goldbach theorem (Helfgott)

Use the Hardy–Littlewood circle method: split the estimate for the number of ways to write $n$ as $p_1+p_2+p_3$ into a 'major arc' contribution (near rationals with small denominator, governed by an explicit singular series) and a 'minor arc' contribution (which must be shown small). Helfgott's advance (2013, expanded 2015) combined sharper minor-arc bounds with precise major-arc estimates and large-scale numerical verification to cover every odd $n$ above an explicit threshold, with smaller cases checked directly by computer. Because of the scale and intricacy of the argument, the complete manuscript underwent an unusually long, multi-round refereeing process before being accepted as a research monograph in the Annals of Mathematics Studies series.