od5 =
od1
Observation deck 1
(Abbr. as "od1")
\documentclass[12pt]{article}
\begin{document}
$$
od1 = (d1)^2 - n
$$
\end{document}
+ od2
Observation deck 2
(Abbr. as "od2")
\documentclass[12pt]{article}
\begin{document}
$$
od2 = (d2)^2 - n
$$
\end{document}
+ od3
Observation deck 3
(Abbr. as "od3")
\documentclass[12pt]{article}
\begin{document}
$$
od3 = od2 - od1
$$
\end{document}
+ od4
Observation deck 4
(Abbr. as "od4")
\documentclass[12pt]{article}
\begin{document}
$$
od4 = od1 + od2
$$
\end{document}
\documentclass[12pt]{article}
\begin{document}
$$
\\ \\ \\ \\
$$
$$
\Delta sieve_{_{od_{_5}}} = \Delta^2 + \dfrac{3}{4} v_1^2, \text{ for } v_1=4k \text{, } k = 2j+1 \text{, for j} \in \mathbb{N}_0
$$
\end{document}
id
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
Index
Used in referring to table rows
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
p
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
Odd natural numbers
q
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
q = p + △
n=pxq
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
Multiply p and q to get n
floor(sqrt(n))
\documentclass[12pt]{article}
\begin{document}
$$
\ \ \text{(} \lfloor \sqrt{n} \rfloor \text{)}
$$
\end{document}
Floor value of sqrt(n)
\documentclass[12pt]{article}
\begin{document}
$$
\ \ \ \ \ \ \ \ \text{(} \lfloor \sqrt{n} \rfloor \text{)}
$$
\end{document}
d1
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
Deck 1 (d1)
\documentclass[12pt]{article}
\begin{document}
$$
d1 = \begin{cases} \lfloor \sqrt{n} \rfloor + a1, & \text{for } \lfloor \sqrt{n} \rfloor = 2k, k \in \mathbb{N} \\ \\
\lfloor \sqrt{n} \rfloor + a2, & \text{for } \lfloor \sqrt{n} \rfloor = 2k + 1, k \in \mathbb{N_0} \ \ \end{cases}
$$
\end{document}
(We had to call these 'type' of columns something, we decided on - "Deck" as the prefix. This column is abbr. as "d1")
d2
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
Deck 2 (d2)
\documentclass[12pt]{article}
\begin{document}
$$
d2 = \begin{cases} \lfloor \sqrt{n} \rfloor + a1 + v1, & \text{for } \lfloor \sqrt{n} \rfloor = 2k, k \in \mathbb{N} \\ \\
\lfloor \sqrt{n} \rfloor + a2 + v2, & \text{for } \lfloor \sqrt{n} \rfloor = 2k + 1, k \in \mathbb{N_0} \text{\ \ } \end{cases}
$$
\end{document}
(We had to call these 'type' of columns something, we decided on - "Deck" as the prefix. This column is abbr. as "d2")
d1 x d1
\documentclass[12pt]{article}
\begin{document}
$$
\ (d1)^2
$$
\end{document}
Deck 1 - Squared
\documentclass[12pt]{article}
\begin{document}
$$
\ \ \ \ \ \ \ \ \ (d1)^2
$$
\end{document}
d2 x d2
\documentclass[12pt]{article}
\begin{document}
$$
\ (d2)^2
$$
\end{document}
Deck 2 - Squared
\documentclass[12pt]{article}
\begin{document}
$$
\ \ \ \ \ \ \ \ \ (d2)^2
$$
\end{document}
od1
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
Observation deck 1 (od1)
\documentclass[12pt]{article}
\begin{document}
$$
\ \ od1 = (d1)^2 - n
$$
\end{document}
(We had to call these 'type' of columns something, we decided on - "Observation deck" as the prefix. This column is abbr. as "od1")
df1
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
Difference between consecutive od1 values
\documentclass[12pt]{article}
\begin{document}
$$
\ \ df1 = od1_{id-1} - od1_{id}
$$
\end{document}
(We had to call these 'type' of columns something, we decided on - "df" as the prefix, which means "difference". This column is abbr. as "df1")
od2
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
Observation deck 2 (od2)
\documentclass[12pt]{article}
\begin{document}
$$
\ \ \ od2 = (d2)^2 - n
$$
\end{document}
(We had to call these 'type' of columns something, we decided on - "Observation deck" as the prefix. This column is abbr. as "od2")
df2
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
Difference between consecutive od2 values
\documentclass[12pt]{article}
\begin{document}
$$
\ \ df2 = od2_{id-1} - od2_{id}
$$
\end{document}
(We had to call these 'type' of columns something, we decided on - "df" as the prefix, which means "difference". This column is abbr. as "df2")
od3
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
Observation deck 3 (od3)
\documentclass[12pt]{article}
\begin{document}
$$
\ \ \ od3 = od2 - od1
$$
\end{document}
(We had to call these 'type' of columns something, we decided on - "Observation deck" as the prefix. This column is abbr. as "od3")
df3
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
Difference between consecutive od3 values
\documentclass[12pt]{article}
\begin{document}
$$
\ \ df3 = od3_{id-1} - od3_{id}
$$
\end{document}
(We had to call these 'type' of columns something, we decided on - "df" as the prefix, which means "difference". This column is abbr. as "df3")
od4
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
Observation deck 4 (od4)
\documentclass[12pt]{article}
\begin{document}
$$
\ \ \ od4 = od1 + od2
$$
\end{document}
(We had to call these 'type' of columns something, we decided on - "Observation deck" as the prefix. This column is abbr. as "od4")
df4
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
Difference between consecutive od4 values
\documentclass[12pt]{article}
\begin{document}
$$
\ \ df4 = od4_{id-1} - od4_{id}
$$
\end{document}
(We had to call these 'type' of columns something, we decided on - "df" as the prefix, which means "difference". This column is abbr. as "df4")
od5
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
Observation deck 5 (od5)
\documentclass[12pt]{article}
\begin{document}
$$
od5 = od1 + od2 + od3 + od4
$$
\end{document}
(We had to call these 'type' of columns something, we decided on - "Observation deck" as the prefix. This column is abbr. as "od5")
df_sum
\documentclass[12pt]{article}
\begin{document}
$$
\ \\
$$
\end{document}
Sum of relevant df columns
\documentclass[12pt]{article}
\begin{document}
$$
df\_sum = df1 + df2 + df3 + df4
$$
\end{document}
(We had to call these 'type' of columns something, we decided on - "df" as the prefix, which means "difference". This column is abbr. as "df_sum")
Dial Settings
a1
a1
Control Variable
\documentclass[11pt]{article}
\begin{document}
$$
\text{if } \Delta = 4k, k \in \mathbb{N}; a1 = -2
$$
$$
\text{if } \Delta = 4k + 2, k \in \mathbb{N}; a1 = -1
$$
$$
\text {(Note: a1 can be in a} \\
\text{complex domain. For explanation } \\
\text{simplicity, we will keep } a1 \in \mathbb{Z_0} \text{)}
$$
\end{document}
v1
v1
Independent Variable
\documentclass[11pt]{article}
\begin{document}
$$
v_1=4k \text{, } k = 2j+1 \text{, for j} \in \mathbb{N}_0
$$
$$
\text {(Note: v1 can be in a} \\
\text{complex domain. For explanation } \\
\text{simplicity, we will observe} \\
\text {the above form and domain,} \\
\text {with v1=v2)}
$$
\end{document}
a2
a2
Control Variable
\documentclass[11pt]{article}
\begin{document}
$$
\text{if } \Delta = 4k, k \in \mathbb{N}; a2 = -1
$$
$$
\text{if } \Delta = 4k + 2, k \in \mathbb{N}; a2 = -2
$$
$$
\text {(Note: a2 can be in a} \\
\text{complex domain. For explanation } \\
\text{simplicity, we will keep } a2 \in \mathbb{Z_0} \text{)}
$$
\end{document}
v2
v2
Independent Variable
\documentclass[11pt]{article}
\begin{document}
$$
v_2=4k \text{, } k = 2j+1 \text{, for j} \in \mathbb{N}_0
$$
$$
\text {(Note: v2 can be in a} \\
\text{complex domain. For explanation } \\
\text{simplicity, we will observe} \\
\text {the above form and domain,} \\
\text {with v1=v2)}
$$
\end{document}
v1 += 8