Equation 677 Database

Magma e1ee09afca92…

magma e1ee09afca92
Size
95
Isomorphism class hash
e1ee09afca9239b348e61c42ba3dc9e8df04fdf85ef90076d422972a4ab3f69e
Satisfies Equation 255
yes
Right-cancellative
yes
Idempotent
no
Submitted by
dwrensha
Submitted at
2026-06-10 12:53:53
Display reorder
94,86,78,75,87,85,84,80,79,81,77,83,76,72,89,74,88,82,73,93,44,12,1,49,40,36,57,16,61,9,69,5,20,24,32,53,65,28,90,56,25,13,60,52,48,68,29,37,21,45,17,33,0,8,64,41,4,91,10,46,70,14,6,2,22,50,26,42,34,38,54,58,66,18,30,62,92,39,7,31,43,71,67,51,11,55,3,63,35,15,19,27,47,59,23 history
Raw table
canonical order · displayed order
Equational Theories
Finite Magma Explorer

Commentary

Size-95 twisted fiber-bundle magma satisfying Eq 677 (and Eq 255). Carrier Z/5 × F_19 with operation (x, s) ◇ (y, t) = ( x + 4·(y − x) mod 5 , α_d·s + β_d·t in F_19 ), where d = y − x mod 5. The base coefficient c = 4 is a primitive 10th root of unity mod 5 (Φ₁₀(4) = 4⁴ − 4³ + 4² − 4 + 1 ≡ 0 mod 5), which is exactly what Eq 677 forces on the base operation x + c·(y − x). The fiber field is F_19 = Z/19Z (prime field). The fiber coefficient pair (α_d, β_d) depends on the base difference d: d = 0: (α, β) = (7, 4) d = 1: (α, β) = (7, 9) d = 2: (α, β) = (11, 15) d = 3: (α, β) = (11, 16) d = 4: (α, β) = (7, 16) Eq 677 holds iff the pairs satisfy, for every d (writing e₁ = −d, e₂ = cd, e₃ = d(1 − c + c²), e₄ = −d/c, all mod 5): β_e₄·α_e₃ + β_e₄·β_e₃·α_e₂·β_e₁ = 1 and α_e₄ + β_e₄·β_e₃·(α_e₂·α_e₁ + β_e₂) = 0, with the d = 0 pair therefore satisfying the standard linear-677 conditions α₀β₀(1 + β₀²) = 1, α₀ + α₀²β₀² + β₀³ = 0. This family strictly generalizes the quadratic-residue-class bundles searched earlier (where (α_d, β_d) was constant on QR/non-QR classes): here the pair may vary with d arbitrarily. Found June 2026 by an exhaustive constraint solve over all per-d coefficient assignments; Eq 677 verified directly on the full Cayley table. The magma is right-cancellative; not idempotent; it satisfies Eq 255 (consistent with the 677 ⇒ 255 finite conjecture). [text written by Claude]

last edited by dwrensha at 2026-06-10 12:53:54 · history