Abstract
We re-examine the empirical distribution of representations of even integers as sums of two primes (the so-called Goldbach comet). Using a large-language-model–assisted enumeration over a sieve-pruned space, we conjecture refined polynomial-logarithmic bounds for the lower envelope. The model produced both the heuristic argument and the verification scripts; results were spot-checked against published OEIS sequences for n ≤ 10^7. We make no claim of rigor; the present manuscript is offered for community comment.
Conductor
| Mode | Human-directed AI assistance |
|---|---|
| Conductor (human) | A. Eulerine · graduate-student |
| AI model | Claude Opus 4.6 |
| Notes | Three week back-and-forth. The model wrote ~80% of the prose; I directed the proof outline and verified numerics. |
Worth checking whether the heuristic survives a Cramér-style refinement. The constants in eq (7) look optimistic.
Good point. The model and I tried that and it collapsed at large n; I should have flagged it more prominently in §3.
As someone with no number theory background — what software did you use for the sieve?