No magma of size 10 satisfies Equation 677.
Jose Brox confirmed this with an exhaustive Mace4 search taking 39 hours (Zulip, Jan 12 2025; configuration: selection_order 2, selection_measure 3, skolems_last). Later independently confirmed using CaDiCaL (a modern CDCL SAT solver) in about 6 hours.
Combined with the same conclusion at sizes 2, 3, 4, 6, and 8 (Tencer, Brox), this completes the proof that no Eq 677 magma exists at any size in {2, 3, 4, 6, 8, 10}.
Sizes 12 and 14 are OPEN: a CaDiCaL SAT attack ran for ~1 year on size 12 without terminating, and a separate experiment covered sizes 12–16 for 244 days without termination. So while 16 is the smallest size > 1 currently KNOWN to admit an Eq 677 magma (e.g. linear over F_16 = GF(2⁴) — Φ_10 has primitive 10th roots there since 10 | 15), 12 and 14 remain undecided.
[text written by Claude]
dwrensha · 2026-05-15 03:26:48
No magma of size 10 satisfies Equation 677.
Jose Brox confirmed this with an exhaustive Mace4 search taking 39 hours (Zulip, Jan 12 2025; configuration: selection_order 2, selection_measure 3, skolems_last). Later independently confirmed using CaDiCaL (a modern CDCL SAT solver) in about 6 hours.
Combined with the same conclusion at sizes 2, 3, 4, 6, and 8 (Tencer, Brox), this means every even size ≤ 14 admits no Eq 677 magma. The smallest even size known to admit one is 16 (e.g. linear over F_16 = GF(2⁴) — Φ_10 has primitive 10th roots there since 10 | 15).
[text written by Claude]
dwrensha · 2026-05-15 03:28:21
dwrensha · 2026-05-15 03:26:48