Control Variables

These need to be optimized for different observation decks, where △ needs to be sieved. Control variables are in the complex domain. For explanation simplicity we have kept them in the integer domain

a1

a1 is a control variable and one of the optimizations is given below when we are sieving the △ in od4
\documentclass[11pt]{article} \begin{document} $$  a1 = \begin{cases} \ \ \ \ 0, & \text{for } \Delta = 4k, k \in \mathbb{N} \\ \\ \ \ -1, & \text{for } \Delta = 4k+2, k \in \mathbb{N} \end{cases} $$ \end{document}

a2

a2 is a control variable and one of the optimizations is given below when we are sieving the △ in od4
\documentclass[11pt]{article} \begin{document} $$ a2 = \begin{cases} \ \ -1, & \text{for } \Delta = 4k, k \in \mathbb{N} \\ \\ \ \ \ \ 0, & \text{for } \Delta = 4k+2, k \in \mathbb{N} \end{cases} $$ \end{document}

Deck

Number of decks are determined based on:
\documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor (mod \text{ } x), \text{ for } x \in \mathbb{N} $$ \end{document}
We have only focussed on           \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor (mod \text{ } 2) $$ \end{document}      on the site.

d1 (abbr. for Deck 1)

\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}

d2 (abbr. for Deck 2)

\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}

Delta Sieve Zone

The region or zone where △ is sieved

Dial Settings

Together, Control and Independent variables are referred as "Dial Settings"

Independent Variables

These need to be optimized for different observation decks, where △ needs to be sieved

v1

v1 is an independent variable and one of the optimizations is given below when we are sieving the △ in od4
\documentclass[11pt]{article} \begin{document} $$ v1 = 4k + 2, k \in \mathbb{N_0} $$ \end{document}

v2

v2 is an independent variable and one of the optimizations is given below when we are sieving the △ in od4
\documentclass[11pt]{article} \begin{document} $$ v2 = 4k + 2, k \in \mathbb{N_0} $$ \end{document}

Observation Deck (od)

This is just a table column, since there are so many of them, making references in Excel style (Column A,B,C, etc.) was not very convenient.
Also, we could not find a suitable close match from existing terms, so we took the liberty to introduce a new one and called it
"Observation Deck (od)" ... inspired from our love and respect for the sailors and explorers of past, current and future - in sea, space and everywhere in between. There are ∞ observation decks.

Each of the unique observation decks are abbr. as od1, od2, od3, od4, od5, od6, ... od∞

od1

\documentclass[12pt]{article} \begin{document} $$ od1 = (d1)^2 - n $$ \end{document}

od2

\documentclass[12pt]{article} \begin{document} $$ od2 = (d2)^2 - n $$ \end{document}

od3

\documentclass[12pt]{article} \begin{document} $$ od3 = od2 - od1 $$ \end{document}

od4

\documentclass[12pt]{article} \begin{document} $$ od4 = od2 + od1 $$ \end{document}

od5

\documentclass[12pt]{article} \begin{document} $$ od5 = od1 + od2 + od3 + od4 $$ \end{document}

od6

We have not introduced od6 yet on the site, but below will be useful for future reference
\documentclass[12pt]{article} \begin{document} $$ od6 = \sqrt{(n \times v1^2) + (od1 \times od2)} \text{ for } v1 = v_2 $$ \end{document}

Steady State Value abbr. as "ssv"

When the respective observation deck value doesn't change and continues until ∞, despite changing p and q, we refer to such state as
"Steady State Value (ssv)"