Diagonal residue families modulo eight

The units of ℤ/8 are 1, 3, 5, 7, each squaring to 1. Fix a unit y. Suppose two families dᵢ and vᵢ of the same nonzero length N have residues in {y, 5y} and {-y, -5y} respectively. Their products and sums agree only when N ≥ 4. A unit diagonal of a dyadic form is read modulo 8 through its product (the determinant square class) and its sum (the oddity). So this counting statement shows that a target diagonal whose entries all have the opposite residue modulo 4 forces rank at least four.

Main results

• ZMod.four_le_of_forall_eq_or_eq: nonempty families in {y, 5y} and {-y, -5y} modulo 8 with the same product and the same sum have at least four members.

theorem ZMod.four_le_of_forall_eq_or_eq {N : ℕ} (hN : N ≠ 0) {y : ZMod (2 ^ 3)} (hy : y * y = 1) {d v : Fin N → ZMod (2 ^ 3)} (hd : ∀ (i : Fin N), d i = y ∨ d i = 5 * y) (hv : ∀ (i : Fin N), v i = -y ∨ v i = -(5 * y)) (hP : ∏ i : Fin N, d i = ∏ i : Fin N, v i) (hS : ∑ i : Fin N, d i = ∑ i : Fin N, v i) : 4 ≤ N

The determinant class and oddity of a diagonal determine at least four entries. Let the residues dᵢ all lie in {y, 5y} and the residues vᵢ all lie in {-y, -5y} modulo 8, for a nonzero number N of indices. If the two families have the same product and the same sum, then there are at least four of them.