The Art of Computer Programming, Volume 2: Seminumerical Algorithms, 3rd Edition

Section 3.3.4

1.  For generators of maximum period, the 1-D accuracy ν₁ is always m, and μ₁ = 2.

2.  Let V be the matrix whose rows are V₁, . . . , V_t. To minimize Y · Y, subject to the condition that Y ≠ (0, . . . , 0) and V Y is an integer column vector X, is equivalent to minimizing (V⁻¹X) · (V⁻¹X), subject to the condition that X is a nonzero integer column vector. The columns of V⁻¹ are U₁, . . . , U_t.

3.  a² ≡ 2a−1 and a³ ≡ 3a−2 (modulo m). By considering all short solutions of (15), we find that $v_3^2=6$ and $v_4^2=4$, for the respective vectors (1, –2, 1) and (1, –1, –1, 1), except in the following cases:

$\begin{array}{l}m=9,a=4\text{ or}7,v_2^2=v_3^2=5;\\ m=9q,a=3q+1\text{ or}6q+1,v_4^2=2.\end{array}$

4.  (a) The unique choice for (x₁, x₂) is $\frac{1}{m}(y_1{u}_{22}-y_2{u}_{21},-y_1{u}_{12}+y_2{u}_{11})$, and this is ...