Vinogradov's theorem

Every sufficiently large odd integer is the sum of three primes. Ivan Vinogradov proved this in 1937 using the Hardy–Littlewood circle method.

Hardy–Littlewood circle method: An analytic number theory method that estimates integer representations by dividing the unit circle into major and minor arcs. Vinogradov used major-arc approximations and minor-arc bounds to count representations by three primes.
Ivan Vinogradov: A Soviet mathematician known for major contributions to analytic number theory. He proved the theorem in 1937 by developing estimates for sums over primes.
Sufficiently large: A mathematical phrase meaning that a claim holds for every input above some fixed threshold. The theorem guarantees representations only beyond a threshold, not for every odd integer.
Weak Goldbach conjecture: The claim that every odd integer greater than five is the sum of three primes. Vinogradov proved this conjecture for sufficiently large odd integers, but not initially for all cases.
Exponential sum: A finite sum of complex exponentials whose phases encode arithmetic information. Prime exponential sums are the central quantities estimated in Vinogradov's proof.
Goldbach conjecture: The conjecture that every even integer greater than two is the sum of two primes. Vinogradov's theorem proves the analogous three-prime statement for sufficiently large odd integers.
Odd number: An integer not divisible by two. The parity of the target integer matches the sum of three odd primes.
Helfgott's proof of the weak Goldbach conjecture: Harald Helfgott's proof that every odd integer greater than five is a sum of three primes. It combined explicit analytic bounds and computation to extend Vinogradov's result to every required odd integer.
Major arcs: Regions near rational points on the unit circle that contribute the main terms in circle-method estimates. Their prime-sum approximations produce the expected positive count of three-prime representations.
Ternary Goldbach problem: The problem of expressing every odd integer greater than five as a sum of three primes. Vinogradov established this claim for all sufficiently large odd integers.

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Goldbach–Euler theoremRelated: It is a major proved result for the related three-prime problem.