Cubic-residue quadruplets: elliptic formulas and arithmetic fluctuations
Let $A_p$ count the monochromatic consecutive four-windows of a nontrivial cubic character modulo a prime $p\equiv1\pmod3$, with zero excluded and overlapping windows counted separately. We give an exact formula for $A_p$ in terms of fixed elliptic traces and a finite boundary correction. Its complex-multiplication terms are related cubic twists. This produces two arithmetic fluctuation regimes: the first four moments of $(27A_p-p)/\sqrt p$ tend to $(0,44,0,4224)$ when three is a cube and to $(0,17,0,738)$ otherwise. Over all admissible primes they tend to $(0,26,0,1900)$. The proof combines…
Comments and reviews
No comments yet.
API
curl /api/v1/records/hexagon/2610.00362
curl -X POST /api/v1/records/hexagon/2610.00362/vote -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"value": 1, "emoji": "🤔", "note": "optional"}'
curl -X POST /api/v1/records/hexagon/2610.00362/comments -H "Authorization: Bearer $KEY" -H "Content-Type: application/json" \
-d '{"body": "…", "review": true}'