The ternary Goldbach conjecture is true
By Harald Andrés Helfgott. 2013
127 citations recorded by the source corpus.
The ternary Goldbach conjecture, or three-primes problem, asserts that every odd integer $n$ greater than $5$ is the sum of three primes. The present paper proves this conjecture. Both the ternary Goldbach conjecture and the binary, or strong, Goldbach conjecture had their origin in an exchange of letters between Euler and Goldbach in 1742. We will follow an approach based on the circle method, the large sieve and exponential sums. Some ideas coming from Hardy, Littlewood and Vinogradov are reinterpreted from a modern perspective. While all work here has to be explicit, the focus is on qualitative gains. The improved estimates on exponential sums are proven in the author's papers on major and minor arcs for Goldbach's problem. One of the highlights of the present paper is an optimized large sieve for primes. Its ideas get reapplied to the circle method to give an improved estimate for the minor-arc integral.
View Paper: https://www.semanticscholar.org/paper/9aa9ba73a52d31f7fd857b4ddda965be66131d7a
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- Harald Andrés Helfgott
- P. P. Varj'u
- Emmanuel Breuillard
- Jean Bourgain
- Ilya D. Shkredov
- Ben Joseph Green
- Alex Gamburd
- L. Pierce
Citations in the corpus
- Minor arcs for Goldbach's problem (2012, methodology): “ wxof Sf,ϕ1(wx) is always large. We will use ϕ 1(t) = t2e−t 2/2, ϕ 2(t) = η 1 ∗Mη 1, where η 1 is 2 times the characteristic function of the interval [1/2,1]. This is the example that will be used in [Helc]. The motivation for the choice of ϕ 1 and ϕ 2 is clear: we have just got bounds based on ϕ 1(”