The Closest Known Attempted Proof of the Twin Prime Conjecture

Source abstract

In Part 1, using a primitive formulation of the circle method, we present a sufficiency condition for the twin prime conjecture that misses by just an epsilon. In Part 2, using the standard formulation of the circle method for the twin prime conjecture, we show that the Hardy-Littlewood minor arc sufficiency condition implies the m-prime conjecture for every even integer m. We show exactly where the proof breaks down, when m is an odd integer.