{"id":"palomar/PALOMAR-2026-11-02-000009","url":"https://github.com/jkfarland-bot/Quartersumladder/tree/756129d46275e42fb4b27b833d156f2a025a2c21","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-11-02-000009-v1.json","repo":"https://github.com/jkfarland-bot/Quartersumladder","tags":["math.NT","msc:11R29","msc:11B68","msc:11A15","msc:11R16","msc:68V20"],"score":0,"title":"QuarterSumLadder: the quarter-sum ladder and the weight-two congruences","trust":"high","domain":"math.NT","source":"palomar","authors":["Joseph S. Farland"],"reviews":0,"subject":"math.NT","summary":"For a prime p = 1 (mod 8) and a quartic character chi modulo p (even, with w = chi(2) = +-1), the development proves that the coordinates of B_(2,chi) = c0 + i c1 are integers with c0 = 4(E_1 + E_-1), c1 = 4(E_i + E_-i) and c0 + c1 = p - 1 modulo 8, and that 8 divides both exactly when w = 1, i.e. when 2 is a fourth power. A Gauss-Lehmer product argument gives the quartic Gauss lemma w = (-1)^(R_1 + R_3) for the quarter counts R_r, the parity of the four quarter counts when w = 1, and the parity of R_2 through the order of 2. For the quarter sum Omega = x + iy (2N(Omega) = h^-(K) for the…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-11-02","reactions":{},"formalized":true,"formalizes":[{"title":"The quarter-sum ladder and the weight-two congruences, formalized in Lean","identifier":"","relationship":"formalizes"},{"title":"The even quartic class: weight-two congruences, the quarter-sum ladder, and 2-primary class groups","identifier":"","relationship":"formalizes"}]}