On Ternary Representations of Odd Integers as Sums of Twin Primes: A Sun-Type Conjecture, a Shifted-Base Refinement, and Numerical Evidence

Abstract

We study the representation of odd integers as sums of three twin primes, in the spirit of Vinogradov/Helfgott's ternary Goldbach theorem and of Zhi-Wei Sun's conjectures on primes and figurate numbers (arXiv:0803.3737). We conjecture that every odd integer N > 9 is a sum of three twin primes, and verify this exhaustively (sieve + FFT-based sumset computation) without exception up to N = 3x10^6. We then define, for each twin prime T, the shifted base of twin primes >= T, and study S(T), the smallest odd threshold beyond which every larger odd integer is representable using only that shifted base. Numerical data for thresholds up to T ~ 2.4x10^5 show that S(T)/3T converges toward 1, with S(T)-3T growing like (log T)^3. We derive this scaling from a Hardy-Littlewood circle-method heuristic, including an explicit computation of the relevant singular series, and find reasonable quantitative agreement with the data. All statements presented are numerical conjectures, not theorems. In particular, none of the material here bears on the twin prime conjecture (the infinitude of twin primes) itself; a proof-by-contradiction attempt to derive that infinitude from the threshold conjecture is explicitly discussed and shown not to close the logical gap. This deposit establishes priority for the specific formulation and numerical results presented (the base conjecture, the shifted-base threshold family S(T), and the singular-series-corrected heuristic comparison). Comments, corrections, and pointers to related or prior work are very welcome.