All-odd ternary Goldbach for $n \ge e^{3100}$

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WeakGoldbach.three_odd_primes_ge_exp3100

by jjosh · Sep 11, 2026 · Mathlib 0df444a (Lean v4.33.1)

goldbachnumber-theoryprimes

For every odd natural number $n \ge e^{3100}$ there exist odd primes $p, q, r$ with $n = p + q + r$ .

This is the Liu–Wang effective-Vinogradov regime, strengthened to odd summands: their estimate shows the weighted representation count is positive (in fact $\gg n^2$ ), while representations involving the prime $2$ are of the shape $\{2,2,n-4\}$ and contribute only $O(\log^2 n)$ , so an all-odd representation must exist.

Preamble

import Mathlib

Formal statement

namespace WeakGoldbach

theorem three_odd_primes_ge_exp3100 (n : ℕ)
    (hn : Real.exp 3100 ≤ (n : ℝ)) (hodd : Odd n) :
    ∃ p q r : ℕ,
      Nat.Prime p ∧ Nat.Prime q ∧ Nat.Prime r ∧
      Odd p ∧ Odd q ∧ Odd r ∧ n = p + q + r := by
  sorry

end WeakGoldbach

Source

M.-C. Liu and T. Wang, On the Vinogradov bound in the three primes Goldbach conjecture, Acta Arith. 105 (2002), strengthened to odd summands by the weighted-count argument

View graph

All-odd ternary Goldbach for n≥e3100n \ge e^{3100}n≥e3100

All-odd ternary Goldbach for $n \ge e^{3100}$