Monogenic cyclic-affine Eq677 magma of order 721. On Z/721, in construction labels, x*y = 46x + 162y + 1 (mod 721).
Element 0 generates the whole magma; there are 612 generators and 1 idempotent element(s). For any element r, the generated submagma has size 721/gcd(721, (46+162-1)r+1). Both coefficients are units modulo 721, so this is a quasigroup (left- and right-cancellative); its fiber matrix is all ones.
The affine Eq677 conditions are ab(1+b^2)=1, a+a^2*b^2+b^3=0, and (b^2(a+1)+b+1)c=0 modulo n. These hold for the displayed parameters. Independently checked all 519841 Eq677 instances, all 721 Eq255 instances, and full set closure from generator 0 directly in the saved table. This is not a counterexample to Eq677 => Eq255.
Found while enumerating monogenic cyclic-affine models of orders below 1000, working upward in order. These are standard affine constructions, added to fill gaps in the database. Construction labels may differ from the server canonical labels. Computed with OpenAI Codex working with dwrensha, 2026-09-27.
dwrensha · 2026-09-27 18:49:34