{"id":"palomar/PALOMAR-2026-10-29-000013","url":"https://github.com/jkfarland-bot/QuadraticFloor/tree/7e51a19db5a27cf064bf5d2f8845812fe1070077","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-29-000013-v1.json","repo":"https://github.com/jkfarland-bot/QuadraticFloor","tags":["math.NT","msc:11A15","msc:11B68","msc:11R29","msc:68V20"],"score":0,"title":"QuadraticFloor: quadratic floor sums and the trace of a quartic Bernoulli number","trust":"high","domain":"math.NT","source":"palomar","authors":["Joseph S. Farland"],"reviews":0,"subject":"math.NT","summary":"For an odd prime p and integers A, B, C, the alternating floor sum S_{A,B,C}(p) is the sum over 0 <= n < p of (-1)^floor((A n^2 + B n + C)/p). The development proves the root-count reduction of S_{A,B,C}(p) to a sum of Legendre symbols over an arithmetic progression when A + B is even, and, for a positive odd A, an odd prime p not dividing A and an admissible pair (B odd, prime to A, 1 <= B^2 - 4AC < 4A), the square-class decomposition: S_{A,B,C}(p) equals (-1)^C minus the average over the Dirichlet characters psi modulo A of conj(psi(B))^2 times the first Bernoulli number, at modulus 8Ap,…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-29","reactions":{},"formalized":true,"formalizes":[{"title":"Quadratic floor sums and the trace of a quartic Bernoulli number, formalized in Lean","identifier":"https://doi.org/10.5281/zenodo.23752228","relationship":"formalizes"},{"title":"Quadratic Floor Sums, Class Numbers, and the Trace of a Quartic Bernoulli Number","identifier":"","relationship":"formalizes"}]}