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\frac{iY * (iY 1)}{2} - iY \leq iX \leq \frac{iY * (iY 1)}{2} - 1
\binom{n}{k} = \binom{n-1}{k}   \binom{n-1}{k-1}
&&x = \sqrt{D}, \\
&&p_0 = 0, \\
&&q_0 = 1, \\
&&a_k =\lfloor \frac{x   p_k}{q_k} \rfloor, \;\; k \ge 0, \\
&&p_{k 1} = a_k Q_k - p_k, \;\; k \ge 0, \\
&&q_{k 1} = \frac{D  - p_{k   1}^2}{q_k}, \;\; k \ge 0.
&& b_j \oplus b_i^{-1} \\
&& = (a_0 \oplus \cdots \oplus a_j) \oplus (a_0 \oplus \cdots \oplus a_i)^{-1} \\
&& = (a_0 \oplus \cdots \oplus a_j) \oplus (a_i^{-1} \oplus \cdots \oplus a_0^{-1}) \\
&& = (a_0 \oplus a_0^{-1}) \oplus \cdots \oplus (a_i \oplus a_i^{-1}) \oplus (a_{i 1} \oplus \cdots \oplus a_j) \\
&& = a_{i 1} \oplus \cdots \oplus a_j
X_N^{(2)} &=& \frac{X_0}{N}R_1 R_2 \cdots R_N   \frac{X_0}{N} R_2 R_3 \cdots R_N   \cdots   \frac{X_0}{N}R_N \\
          &=& \frac{X_0}{N} (R_1 \cdots R_N   R_2 \cdots R_N   \cdots   R_N) \\
          &=& \frac{X_0}{N} \sum_{i=1}^N \prod_{j=i}^N R_j \\
          &\equiv& \frac{X_0}{N} \sum_{i=1}^N {\cal R}_i