Equation 677 Database

Constant-offset linear magma over F_31: x ◇ y = 5x + 27y + 28 (mod 31). Size 31, prime, fully NON-idempotent (no idempotents at all), right-cancellative. |Aut(M)| = 31 (only the Z_31 translation; no multiplicative stabilizer). The operation is x ◇ y = 5x + 27y + 28 (mod 31), or equivalently x ◇ y = x + 27 * (y - x) + 28. Since 5 + 27 = 32 = 1 (mod 31), this falls into the affine-line family x ◇ y = a*x + (1-a)*y + c with a = 5, c = 28. The slope alpha = 27 is one of the four primitive 10th roots of unity in F_31* (the Phi_10 roots being {15, 23, 27, 29}); without the +28 constant offset this would be the same idempotent linear magma magma#543c5479 (which is x ◇ y = 5x + 27y). Diagonal: T(x, x) = (5 + 27) * x + 28 = x + 28 mod 31. So x ◇ x = x + 28, never x: zero idempotents. This is the only non-fully-idempotent "affine line + constant" 677 magma at size 31 in the DB. The closest cousin is magma#3cb2b467 (x ◇ y = 5x + 9y), which is also non-idempotent but uses general (a, b) with a + b = 14 ≠ 1 (one idempotent at x = 0 only). Compare with the 4 fully idempotent linear F_31 magmas in the DB (one per Phi_10 root): magma#7ba82cf6: x ◇ y = 17x + 15y (alpha = 15) magma#543c5479: x ◇ y = 5x + 27y (alpha = 27) magma#fdf14b96: x ◇ y = 9x + 23y (alpha = 23) magma#7ee5b4b8: x ◇ y = 3x + 29y (alpha = 29) Each has |Aut| = 930 = 30 * 31 = |AGL(1, 31)|. This non-idempotent variant has only |Aut| = 31 (translation only) because the constant offset c = 28 breaks the multiplicative symmetry: multiplication by mu ∈ F_31* would satisfy mu * (5x + 27y + 28) = 5(mu x) + 27(mu y) + 28 only if 28*mu = 28, hence mu = 1. Eq 677 verified algebraically: with f(z) = alpha * z + c (f(0) = c, so non-idempotent), the magma x ◇ y = x + f(y - x) satisfies Eq 677 iff alpha is a Phi_10 root in F_p AND c is arbitrary (any c gives Eq 677). The constant c shifts the magma "in unison" without breaking the equation. [text written by Claude]