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We aim to express observation deck

              Observation Deck (od)

                     (Abbr. as "od")

\documentclass[12pt]{article} \begin{document} $$ \text{This is a table column, since there are} \\ \text{so many of them, making references in } \\ \text{Excel style (Column A,B,C, etc.) wasn't} \\ \text{very convenient. Also, we couldn't find} \\ \text{a suitable close match from existing } \\ \text{terms, so we took the liberty to} \\ \text{introduce a new one and called it} \\ \text{"Observation Deck (od)" ... } \\ \text{} \\ \text{... inspired from our love and respect} \\ \text{for the sailors and explorers of past, } \\ \text{current and future - in sea, space and} \\ \text{everywhere in between} \\ \text{} \\ \text{There are ∞ observation decks. Each of} \\ \text{the unique observation decks are abbr.} \\ \text{as od1, od2, od3, od4, od5, od6, ... od∞} \\ $$ \end{document}

values as,
"f(△) + f(independent variables)

Independent Variables

(Abbr. as v1, v2, etc)

 + some constant C", in conjunction with different control

  Control Variables

  (Abbr. as a1, a2, etc)

 variables. Together, the independent and control variables will be referred as "Dial Settings"
We will explain in detail how the values for 16 columns are calculated in the below Table and lay the foundations for △ Sieve

  \documentclass[16pt]{article} \begin{document} $$ \ \ \Delta = |p-q| = 12 $$ \end{document}

id

\documentclass[12pt]{article} \begin{document} $$ \ \\ $$ \end{document}

Index

Used in referring to table rows

 		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")

 		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")

 		df_sum

\documentclass[12pt]{article} \begin{document} $$ \ \\ $$ \end{document}

Sum of relevant df columns

\documentclass[12pt]{article} \begin{document} $$ df\_sum = df1 + df2 $$ \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} $$ \text{if } \Delta = 4k, k \in \mathbb{N}; a1 = 0 $$ $$ \text{if } \Delta = 4k + 2, k \in \mathbb{N}; a1 = -1 $$ $$ \text {(Note: a1 can be in a} \\ \text{complex domain. For explanation } \\ \text{simplicity, we will keep } a1 \in \mathbb{Z_0} \text{)} $$ \end{document}

v1

v1

Independent Variable

\documentclass[11pt]{article} \begin{document} $$ \ \ \ \ \ \ \ \ \ v1 = 4k + 2, k \in \mathbb{N_0} $$ $$ \text {(Note: v1 can be in a} \\ \text{complex domain. For explanation } \\ \text{simplicity, we will observe} \\ \text {the above form and domain,} \\ \text {with v1=v2)} $$ \end{document}

a2

a2

Control Variable

\documentclass[11pt]{article} \begin{document} $$ \text{if } \Delta = 4k, k \in \mathbb{N}; a2 = -1 $$ $$ \text{if } \Delta = 4k + 2, k \in \mathbb{N}; a2 = 0 $$ $$ \text {(Note: a2 can be in a} \\ \text{complex domain. For explanation } \\ \text{simplicity, we will keep } a2 \in \mathbb{Z_0} \text{)} $$ \end{document}

v2

v2

Independent Variable

\documentclass[11pt]{article} \begin{document} $$ \ \ \ \ \ \ \ \ \ v2 = 4k + 2, k \in \mathbb{N_0} $$ $$ \text {(Note: v2 can be in a} \\ \text{complex domain. For explanation } \\ \text{simplicity, we will observe} \\ \text {the above form and domain,} \\ \text {with v1=v2)} $$ \end{document}


From the above Table:

1. Let's see how △ is related with od4 column when △ = 12.
One can also experiment with different even △ values and observe how the sieving will work in od4 & od5

\documentclass[12pt]{article} \begin{document} $$ \Delta sieve_{_{od_{_4}}} = \begin{cases} \text{f(△)} = \dfrac{(\Delta)^2}{2} \\ \\ + \text{ f(independent variables)} = \dfrac{(v_1)^2}{2}, \text{ for } v_1=4k + 2, k \in \mathbb{N}_0 \\ \\ + \text{ Some constant C} = 0 \ \ \end{cases} $$ $$ \text{The above 3 components are summarized as:} \\ $$ $$ \Delta sieve_{_{od_{_4}}} = \dfrac{(\Delta)^2}{2} + \dfrac{(v_1)^2}{2}, \text{ for } v_1=4k + 2 \text{, } k \in \mathbb{N}_0 $$ $$ \text{Note: Although control variables do not feature above, they are } \text{equally important for the observed} \\ \text{behavior and are working behind the scene.} $$ $$ $$ $$ \\ \text{For △ = 12, the first sieve value can be calculated as:} $$ $$ \Delta sieve_{_{od_{_4}}} = \dfrac{(12)^2}{2} + \dfrac{(2)^2}{2} = 74 $$ \end{document}

2. Also, the value of p when v1 = v2 = 2 can also be expressed as f(△)
    Note: The subscript "ssv" stands for "steady state value" from context of

\documentclass[12pt]{article} \begin{document} $$ p_{_{ssv}}= \begin{cases} \left(2 \times \left(\dfrac{\Delta - 4}{4}\right) \left(\dfrac{\Delta - 4}{4} + 1\right)\right)+ 1, & \text{for } \Delta = 4k, k \in \mathbb{N} \\ \\ \left(2 \times \left(\left(\dfrac{\Delta - 6}{4} + 1\right)^2 - 1\right)\right) + 3, & \text{for } \Delta = 4k+2, k \in \mathbb{N} \end{cases} $$ $$ $$ $$ \text{Here, △ = 12, so we will plug this value in equation for △=4k form, to get the value of p} $$ $$ p_{_{ssv}}= \left(2 \times \left(\dfrac{12 - 4}{4}\right) \left(\dfrac{12 - 4}{4} + 1\right)\right)+ 1 \\ \ \ \ \ \ = \left(2 \times 2 \times 3\right) + 1 \\ \\ \\ \ \ \ \ \ = 13 \text{ (the cell filled in blue in the above Table under the p column)} $$ \end{document}

3. Lastly, it is conjectured that for given optimizations of control variables (a1 and a2) and independent variables (v1=v2=2) the od4 value obtained as "f(△) + f(independent variables)" will remain constant until ∞ starting from p, which is also expressed as some other f(△) (as given in point #2 above)

Delta (△ = |p-q|) Column #1: "id" Column #2: "p" Column #3: "q" Column #4: "n" Column #5: "fsqrt" Column #6: "d1" Column #7: "d2" Column #8: "d1 x d1" Column #9: "d2 x d2" Column #10: "od1" Column #11: "df1" Column #12: "od2" Column #13: "df2" Column #14: "od3" Column #15: "od4" Column #16: "df_sum" a1 a2 v1 v2
Delta (△ = |p-q|)
Any positive natural number. While the concepts are applicable to both even and odd natural numbers, we have restricted the △ to be any positive "even" natural number under the experiment (od4 & od5) sections for explanation simplicity. In the above table, we have taken △ = 12

Column #1: "id"
This is just the index column and is useful when referring to any specific row of the Table

Column #2: "p"
Positive odd natural numbers starting from 1

Column #3: "q"
q = p + △ (we study all combination of p and q for a given fixed △)

Column #4: "n"
n = p × q

Column #5: "floor(sqrt(n))"
We take the square root of "n" and apply the floor function. Mathematically, this is als represented as ->    \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor \text{)} $$ \end{document}

Column #6: "d1" (d1 is the abbr. for "Deck 1")
To get the value of d1, we look at    \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor $$ \end{document}  . If    \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor \text{)} $$ \end{document}   is EVEN, we add the first control variable "a1" (more on a1) to    \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor \text{)} $$ \end{document}    and obtain d1 =    \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor $$ \end{document}    + a1.
For e.g. : In the above table, refer the second row (id=2), where   \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor = 6 $$ \end{document}     , as this is even, we add
a1 = 0 and obtain \documentclass[12pt]{article} \begin{document} $$ d1 = 6 + 0 = 6 $$ \end{document}

 Likewise, If    \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor \text{)} $$ \end{document}   is ODD, we add the second control variable "a2" (more on a2) to    \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor \text{)} $$ \end{document}    and obtain
d1 =    \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor $$ \end{document}    + a2.
For e.g. : In the above table, refer the first row (id=1), where   \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor = 3 $$ \end{document}     , as this is odd, we add
a2 = -1 and obtain \documentclass[12pt]{article} \begin{document} $$ d1 = 3 + (-1) = 2 $$ \end{document}

Succinct definition of 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}

Column #7: "d2" (d2 is the abbr. for "Deck 2")
The fundamental principle to calculate d2 is same as that of d1. However, two paths can be taken:
      1) We look at    \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor $$ \end{document}   and check if it is EVEN or ODD - This is the path we have taken; OR
      2) Look at d1 and check if it is EVEN or ODD

 So, If    \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor \text{)} $$ \end{document}   is EVEN, we add the first control variable "a1" (more on a1) and the first independent variable "v1" (more on v1) to    \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor \text{)} $$ \end{document}    and obtain d2 =    \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor $$ \end{document}    + a1 + v1.
For e.g. : In the above table, refer the second row (id=2), where   \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor = 6 $$ \end{document}     , as this is even, we add a1 = 0, v1 = 2 and obtain \documentclass[12pt]{article} \begin{document} $$ d2 = 6 + 0 + 2 = 8 $$ \end{document}

 Likewise, If    \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor \text{)} $$ \end{document}   is ODD, we add the second control variable "a2" (more on a2) and the second independent variable "v2" (more on v2) to    \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor \text{)} $$ \end{document}    and obtain d2 =    \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor $$ \end{document}    + a2 + v2.
For e.g. : In the above table, refer the first row (id=1), where   \documentclass[12pt]{article} \begin{document} $$ \lfloor \sqrt{n} \rfloor = 3 $$ \end{document}     , as this is odd, we add a2 = -1, v2 = 2 and obtain \documentclass[12pt]{article} \begin{document} $$ d2 = 3 + (-1) + 2 = 4 $$ \end{document}

Succinct definition of 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}

Column #8: "d1 x d1"
The value of d1 is squared in this column, i.e.      \documentclass[12pt]{article} \begin{document} $$ (d1)^2 $$ \end{document}

Column #9: "d2 x d2"
The value of d2 is squared in this column, i.e.      \documentclass[12pt]{article} \begin{document} $$ (d2)^2 $$ \end{document}

Column #10: "od1" (od is the abbr. for "Observation Deck" and od1 is the first observation deck)
\documentclass[12pt]{article} \begin{document} $$ \ \ od1 = (d1)^2 - n $$ \end{document}
For e.g. : In the above table, refer the sixth row (id=6), where \documentclass[12pt]{article} \begin{document} $$ \ (d1)^2 = 196 $$ \end{document}     , \documentclass[12pt]{article} \begin{document} $$ \ \ n = 253 $$ \end{document}     , then,
\documentclass[12pt]{article} \begin{document} $$ \ \ \ \ \ \ \ \ \ \ \ od1 = (d1)^2 - n = 196 - 253 = -57 $$ \end{document}

Column #11: "df1" (df is the abbr. for "Difference", and "df1" signifies the difference between the consecutive od1 values)
\documentclass[12pt]{article} \begin{document} $$ \ \ df1 = od1_{id-1} - od1_{id} $$ \end{document}
Here, if     \documentclass[12pt]{article} \begin{document} $$ od1_{id} $$ \end{document}    is the value of od1 at a given table row with some id, then    \documentclass[12pt]{article} \begin{document} $$ od1_{id-1} $$ \end{document}    is the od1 value of the previous row.
For e.g. : In the above table, refer the fourth row (id=4), where od1 = -33, then, the value of od1 for id-1 is the value of od1 at id = 3, which is -21. So, \documentclass[12pt]{article} \begin{document} $$ df1 = od1_{id-1} - od1_{id} = -21 - (-33) = 12 $$ \end{document}

Note: The value of all columns with "df" prefix in the first row, i.e. id=1 is assumed to be 0

Column #12: "od2" (od is the abbr. for "Observation Deck" and od1 is the second observation deck)
\documentclass[12pt]{article} \begin{document} $$ \ \ od2 = (d2)^2 - n $$ \end{document}
For e.g. : In the above table, refer the sixth row (id=6), where \documentclass[12pt]{article} \begin{document} $$ \ (d2)^2 = 256 $$ \end{document}     , \documentclass[12pt]{article} \begin{document} $$ \ \ n = 253 $$ \end{document}     , then,
\documentclass[12pt]{article} \begin{document} $$ \ \ \ od2 = (d2)^2 - n = 256 - 253 = 3 $$ \end{document}

Column #13: "df2" (df is the abbr. for "Difference", and "df2" signifies the difference between the consecutive od2 values)
\documentclass[12pt]{article} \begin{document} $$ \ \ df2 = od2_{id-1} - od2_{id} $$ \end{document}
Here, if    \documentclass[12pt]{article} \begin{document} $$ od2_{id} $$ \end{document}    is the value of od2 at a given table row with some id, then    \documentclass[12pt]{article} \begin{document} $$ od2_{id-1} $$ \end{document}    is the od2 value of the previous row.

For e.g. : In the above table, refer the fourth row (id=4), where od2 = 11, then, the value of od2 for id-1 is the value of od2 at                 id = 3, which is 15. So,   \documentclass[12pt]{article} \begin{document} $$ df2 = od2_{id-1} - od2_{id} = 15 - 11 = 4 $$ \end{document}

Note: The value of all columns with "df" prefix in the first row, i.e. id=1 is assumed to be 0

Column #14: "od3" (od is the abbr. for "Observation Deck" and od3 is the third observation deck)
\documentclass[12pt]{article} \begin{document} $$ od3 = od2 - od1 $$ \end{document}
For e.g. : In the above table, refer the sixth row (id=6), where    \documentclass[12pt]{article} \begin{document} $$ od2 = 3 $$ \end{document}   ,  \documentclass[12pt]{article} \begin{document} $$ od1 = -57 $$ \end{document}    , then,
\documentclass[12pt]{article} \begin{document} $$ \ \ \ \ \ od3 = od2 - od1 = 3 - (-57) = 60 $$ \end{document}

Column #15: "od4" (od is the abbr. for "Observation Deck" and od4 is the fourth observation deck)
\documentclass[12pt]{article} \begin{document} $$ \ \ od4 = od2 + od1 $$ \end{document}
For e.g. : In the above table, refer the sixth row (id=6), where    \documentclass[12pt]{article} \begin{document} $$ od2 = 3 $$ \end{document}   ,   \documentclass[12pt]{article} \begin{document} $$ od1 = -57 $$ \end{document}    , then, \documentclass[12pt]{article} \begin{document} $$ \ \ \ \ \ od4 = od2 + od1 = 3 + (-57) = -54 $$ \end{document}

Column #16: "df_sum"
This is the sum of the "relevant" df columns.

How to identify the relevant df columns?
In the above Table, we are going to sieve △ is od4 column. Now, od4 is constituted from od1 and od2, and hence the relevant df columns will be the corresponding df columns of od1 and od2, which are df1 and df2 respectively.
In the above table:         \documentclass[12pt]{article} \begin{document} $$ df\_sum = df1 + df2 $$ \end{document}        , or can also be represented as              \documentclass[12pt]{article} \begin{document} $$ \sum\limits_{i=1}^{2} {df}_i $$ \end{document}

Why is this important?
This gives △ Sieve its legs and wings. One of the important conjectures is around this property.

△ will be emitted as -> "f(△) + f(independent variables) + some constant C", when:
\documentclass[12pt]{article} \begin{document} $$ \sum\limits_{i=x}^{k} {df}_i = 0 \text{ for x,k } \in \mathbb{N} $$ \end{document} NOTE: There will be trivial zeroes for the above equation, which can be easily identified and ignored.

If the WHY is getting confusing, you can ignore it for now. We thought to put one of the simple forms out there ...
(You guessed it right, there is a complex form also)

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}

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}