Equation 677 Database

No Eq 677 magma of size 15 is currently known; the existence question is OPEN but several substantial partial results restrict where one could live. Linear constructions are ruled out: 15 is not a prime power, so there is no field F_15. The standard linear construction x ◇ y = (1−α)x + αy requires a finite field, and the ring Z/15Z (the only sensible 15-element ring) doesn't help either: (Z/15Z)* ≅ Z/2 × Z/4 ≅ Z/8 by CRT (since |F_3*| · |F_5*| = 2·4), and 10 ∤ 8, so Φ_10 has no roots in Z/15Z. (Also 15 is not a square, ruling out AG(2, q) line magmas, and 15 isn't a prime square so no Zassenhaus exceptional or order-15 Dickson near-field exists.) Translation-invariant magmas of size 15 are provably empty: Jose Brox confirmed (Zulip, Jan 2025) that there are no translation-invariant Eq 677 models of the form x ◇ y = x · f(x⁻¹ y) up to size 17 included, for ANY group structure on the carrier. Since the only group of order 15 is C_15 (cyclic — by gcd(3, 5−1) = 1 and Schur–Zassenhaus, this is the unique group), this exhaustively rules out the translation-invariant case at size 15. General (non-translation-invariant) exhaustive search is inconclusive: a CaDiCaL SAT solver ran for about 6 days searching for a size-15 magma satisfying Eq 677 anti Eq 255, then segfaulted without resolution (Zulip thread; the searcher commented that finding one was 'unlikely'). The pure Eq 677 search at size 15 has not been finished by any tool. So size 15 sits in the OPEN-but-unlikely category alongside sizes 12 and 14: known constructions don't produce one, partial-search results restrict where one could be, but full non-existence has not been proven. [text written by Claude]