Asmak-Tantra (The Number Theory of Aryabhatta)

Abstract: In this text, author translate three verse(Definitions of Aryabhatiya) and one verse

of BrahamSphuthaSidhanta written by Aryabhatta and Brahmagupta respectively. Mostly

this analysis is the investigation of "Number Theory of Aryabhatta". These verse were

translated in past by many wise and capable translators. The available translation of these

verse is based on spirituality, but not on Mathematics. Because these particular verses contain

some "Spiritual Nouns".

But, author believe that " if something is written by Aryabhatta/Brahmagupta then it

must contain much mathematics rather than anything. So, while One translate work of

these brilliant writers then the focus of one, must be on Mathematics rather than

spirituality or anything".

And after translating the script , author find "The Number theory of Aryabhatta"(Asmak-

Tantra).

In this theory Author explain the cast and class of Natural Whole Numbers, on the base of

Theorem of Trinity ( birth of natural numbers). Then I introduce the concept of "The

symmetrical digits" among Natural Numbers and prove that all SD (except digit of 1) are

composite. After that I conjecture that SD of 1( except 11) will be composite?

After that author explain the "strong" version of Goldback Conjecture and find that this

conjecture fallow symmetrical pairing of Primes. So, Author purify the statement of

Goldbach Conjecture (which show symmetry to Riemann Hypothesis).

Technical terms: Newborn Number(N),Numbers Ancestor(G,F),Middle Position(M),

Symmetrical Pairing(SYP), Number of SYP, Alternate Operation, The Great Matrix(of all

Numbers)10× 10, Secondary Matrix(9× 9),Primate Position, Symmetrical Digits,

The Counting, is ancient mathematical practise that always enhance the

knowledge of a mathematician. if Mathematics is "The God", then counting is "The

Prayer". "Mathematician" is "Devotee" and "Numbers" are Mantra`s( words or the

sound of prayer). Many Ancient Mathematicians, who belong to different civilization

appeared at different time , reviled the Truth about Mathematics and continue, The

Counting.

The Sumerians', The Babylonians', The Egyptians', The Greeks', The Chinese , The

Mohammedians' and many more along with The Indians' -All Humankind found their own

concepts, algorithm, theorems. Or in my opinion, Mathematics reviled own self to

Humankind at different time.

But here, the subject is formation of number line in the light of first three element of number

line(1,2,3). "Three", is a special number, it is the third successive element of natural numbers

first odd/ prime in nature. It appears in several written work of mathematicians like

"Aryabhatta" ( 5-6 CE, India)[1,2,3] and "Brahmagupta" ( 7 CE, India)[4,5] , like a

code.

1,2,3,4,5,6,7,8,9,10,11,,, is the natural series with common difference 1.

"Three" is very interesting number since "Human" start counting. There are many culture

that expresses "Divine Power" or "The God /The One" into "Three", like in India, we

"Hindus" believe in trinity "Brahama, Visnu, and Mahesh". They are the deities and main

cause for "Birth, Sustain and Distraction" of "The Universe".

The Christians, also expresses "The One into Trinity" (The Holy Trinity), "Father, Son

And The Holy Spirit". In China, there is a culture, Daoism. they also said that: The First

Dao appeared, Second Dao appeared and then The Third Dao appeared and this keep going,

like One, Two, Three ,,,,,,,,and so on.

In Mathematics and science, the appearance of Three is very frequent. The Time is divided

into Three as "Past, Present and The Future". The Space, is expressed by Euclid is Into

Three as X-Y-Z axis. Newton reviled motion into "Three laws".

Three, is the number who always attracted the Mathematicians and the Theologises both.

.

Here I want that my readers must aware of a fact that Nicola Tesla( a practical physicist)

once wrote to the American Red Cross, New York City, about a strange message from

somewhere outer space, and claim that it massage came from Aliens. The latter is-

To the American Red Cross, New York City

The retrospect is glorious, the prospect is inspiring: Much might be said of both. But

one idea dominates my mind. This — my best, my dearest — is for your noble cause. I

have observed electrical actions, which have appeared inexplicable. Faint and uncertain

though they were, they have given me a deep conviction and foreknowledge, that ere

long all human beings on this globe, as one, will turn their eyes to the firmament above,

with feelings of love and reverence, thrilled by the glad news: “Brethren! We have a

message from another world, unknown and remote. It reads: one… two… three…”

Christmas 1900

Nikola Tesla

Original Latter is-

(Source: Tesla Society)

My intention is not to prove that "Alien"/"God" exist or not. No, but my focus is on that

there was a wise man, a great Scientist and mystical person, who always attracted by Three,

and believe that "1,2,3" is a Code.

While I was reading Aryabhatta, then I find that Natural Number appears with some Laws. In

this text I translate mathematically, the first two verse and an another verse of the text

"Aryabhatiya"( originally in Sanskrit script)[1,2,3] written by Aryabhattya in 5th century.

And first verse of text "BrahmaSphuthaSidhata"(Sanskrit)[4,5] written by Brahmagupta in

7th centurey.

Aryabhatiya written by Aryabhatta has Four parts Dasagitika(stanza of tens),

Ganitam(Mathematics), Kalkiram(Action of Space-time) and Golam( The Sphere).

Method

Section A.

The method for this analysis of Aryabhatta text( in Sanskrit script) about appearance or birth

of natural numbers is, "Partition".

As we know that in "Partition Theory" we divide natural (whole) number into other whole

parts, for example-

7=1+6=1+2+4=1+1+2+2+1

Here we can divide 7 into two parts-1+6 or 2+5 etc.,

Into three parts 1+2+4 or 1+3+3 etc.,

Into four parts 1+1+1+3+1 or 1+2+2+2 etc.

But in this analysis I use Bi-Partition, when a whole is express into Two Parts then this

method is called(here) Bi-Partition. For example 7 can be express into two parts as

7=1+6=2+5=3+4 only.

But before any further discussion, I want that my readers must look at some elementary and

nessesrey definitions to understand analysis in this section.

Definition's:

(A-1) Natural(whole) Numbers and Number Line( or series): The Numbers, which exist

physically in nature. The line form by Natural Whole Numbers in ascending order.

1,2,3,4,5,6,7,8,9,10,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,

(A-2) Birth of Natural( Whole) Number: As we know that Natural Numbers appear on

natural number line, successively. Then The Event, when a whole natural number appear on

number line, is called( here) Birth of Natural Number.

(A-3) Newborn: When numbers are occupying their place on Natural Number Line, then

newborn is that "Dear One" who is consider by Mathematician at the event of the birth. It is

denoted by ℕ or n".

For example let us look at the birth of 5,

1,2,3,4,ℕ

1,2,3,4,5

(A-4) Ancestor Numbers: If a Newborn ℕ get birth on particular event. Then all whole

natural numbers before it, are Ancestors of ℕ. In above example 1,2,3,4 are Ancestors of 5.

(A-5) First Ancestor: If a Newborn ℕ get birth. Then "The Number" just before ℕ, is

called First Ancestor. It is Denoted by FA or 𝑭𝟏.

𝐹1 = ℕ− 1

If we consider birth of 5,then number 4 is first Ancestor for 5.

(A-6) The Great Ancestor : As we know that in the Natural Progression, first Natural

Number 1, get birth. so, it is obvious that it is " The Great Ancestor" for every Natural

Number. It is denoted (here) by 𝐺1.

From definition (5),

𝐹1 = ℕ− 1

𝐺1 = ℕ− 𝐹1 , where 𝐺1 = 1

(A-7) Middle Position: If we are analyzing the birth of natural number N. Then we have to

do numbering/indexing of all Ancestor of ℕ. This numbering of Ancestor will be done about

to Middle Position(M) by two references . One is The first Ancestor(𝑭𝟏) and other is The

Great Ancestor(1). Half of numbering will be done for each reference.

𝑴 =ℕ

𝟐

{ Remark: For Odd Newborn N, half of N will be not a whole number. So, for all Odd we

consider, M= Null }

Example; for birth of 6(Even),

1,2,3,4,5, ℕ

We have five Ancestor, then numbering/indexing about middle(3) is

𝑮𝟏 = 𝟏, 𝑮𝟐 = 𝟐 M=3

𝑭𝟏 = 𝟓, 𝑭𝟐 = 𝟒

for birth of 7(Odd),

1,2,3,4,5,6, ℕ

we have six Ancestor then,

𝑮𝟏 = 𝟏, 𝑮𝟐 = 𝟐, 𝑮𝟑 = 𝟑

𝑭𝟏 = 𝟔, 𝑭𝟐 = 𝟓, 𝑭𝟑 = 𝟒

M = Null

(A-8) Symmetrical Pairing(SYP)/ Number of Pairs

(a) SYP: The one-to-one pairing of Ancestor(G-F) (about Middle position, M) is

called(here), as The Symmetrical pairing(SYP). The order of SYP is denoted by s=1,2,3,4,5

as 𝐺𝑠, 𝑎𝑛𝑑 𝐹𝑠 (𝐺1+𝐹1, 𝐺2 + 𝐹3, 𝐺3 + 𝐹3 ,,,,,,,,,so on).

(b) Number of pairs(s): Number of pairs If

(I) 𝒏 is Even: 𝒔 =𝒏

𝟐

(II) 𝒏 is Odd: 𝒔 = 𝒏−𝟏

𝟐

(A-9) Root/ origin of Natural series:

(a) The origin of unity (1), is zero(0). It is whole number but not natural. So, it will not

appears in this analysis.

(b) The root of natural series is first element of that series, The One or Unity(1). It is the

measurement of existence.

(A-10): Magnitude and Nature:

(a) Magnitude: The magnitude of any whole natural is its size with respect to The Unity(1).

(b) Nature : "Nature of whole natural is that, either it is even or odd". 2 is deciding factor

for classification of whole natural, according to AsmakaSiddhant( Theory of Aryabhatta).

It will be defined further.

Aryabhatiya Verse(1) of Dasagitika section :[1,2]

Translation by Author:

[ The One(Ekm) is True Deity(Brhamin), the creator of all (also Numbers), but he/it is also

Many( Anekam), he/it(The Brahmin) is The lord( Prani-patey) of all(Numbers),.

Then, Aryabhatta, establish(Gadati) "The Trinity" { Three's (1,2,3)} as

(1) Mathematics (Ganitam)

(2) Physics of Space-Time ( Kal-Kriya)

(3) The Sphere (Golam) ]

{ Remark: Word "Aryabhuta-sya-trini" means for "The Trinity of Aryabhatta". The last

word(of this sentence) is for great importance here, "Trini" is "Three" in Sanskrit. }

We can understand above statement by Brahmagupta who say( in his text BrahmaSphuttha-

Siddhanta or "Theory of Brahmagupta" ).

The very first verse[4,5], of his text, is gratitude of Brahmagupta to his Deity(MAHADEVA),

Translation: I pray to "The One", who worshiped by Titans, Demons. "His Glorious Feet" are

(like jewels) shining in "The Morning Light" ( The Morning Light desired to kiss "His Feet" ).

He is only cause for "The Trinity" (Beginning, Sustain, and Destruction) of The Universe. I

pray to The Supreme Being, "MAHADEVA".

Mathematical description of above verse is- Brahmagupta is defining The

One(1)(MAHADEVA), who is deity of all( Demons and Titans)(2) and he is cause of Three

(Birth, Sustain and Destruction)(3).

The one according to Brahmagputa as

(1) is The Deity of Birth (The Absolute, The Creator)

(2) is The Deity of Sustain ( The Maintainer)

(3) is The Deity of Death ( The Destructor)

so, I conclude the two statement together as following and called(here) The Trinity. The

joint statement of Aryabhatta-Brahmagupta as-

(A-1):Theorem of Trinity: The formation of Natural Number Line and the birth of Natural

Numbers:

The formation of Natural Number line or the birth of Natural Numbers, happens according to

"Law of Three( 1,2,3,,,)". The Newborn(Number) will get "The Nature" and "The

Magnitude" By this law.

Law of Three(1,2,3,,,): It state that " The magnitude of Newborn will be equal to the Sum

of SYP( and Even Newborn will be twice of middle Ancestor ). The nature of Newborn will

be opposite of preceding ( First Ancestor)".

...............................................Equation (A-1)

ℕ 𝑜𝑝𝑝.𝑜𝑓 𝐹1⇔ {𝐺1+𝐹1 = 𝐺2 + 𝐹2 = 𝐺3+𝐹3 = ⋯……………𝑠𝑜 𝑜𝑛}

for Even Newborn ℕ = 2 ×𝑀

ℕ = 𝐹𝑠 + 𝐺𝑠 ( Nature is opposite of 𝐹1)

ℕ = 2 ×𝑀 ( For Even Newborn)

s=1,2,3,,,,,,,,,,to 𝑁

2( For even)

s= 1,2,3,,,,,,,, to (𝑁−1)

2 ( for odd)

(A-1) Proof: (Method - addition by partition, lemma by lemma)

Lemma A-1: If there exist 1. (The reason of existence of one is fundamental )

Then we have 1.

Lemma A-2:Event of birth N. We have 1,N

One is added own self produce Two.

1+1=2 is Even

Now we have 1,2,N

Lemma A-3:Event of birth of next

Then 1 add with 2, produce 3

1+2=3 is Odd

Now , we have 1,2,3 .

This is The Trinity of Mathematics. The law of birth of Real Numbers is hidden in between

them, Two Ancestor get fussed for Third(Newborn), that is why I call it Law of Three or

theorem of Trinity.

Lemma A-4:(The Birth of Four) 1,2,3(odd),N

𝐺 = 1 M= 2 𝐹 = 3

New-born N(opposite of Previous)= 𝑮𝟏+𝑭𝟏= 2× M

Then,

N(even) = 1+3 = 2×2 = 4

Now ,we have 1,2,3,4.Then

lemma A-5:(The Birth of Five) 1,2,3,4,N

𝐺 = 1,2 𝑀 = 𝑁𝑢𝑙𝑙 𝐹 = 3,4

N(odd)=𝑮𝟏+𝑭𝟏 = 𝑮𝟐+𝑭𝟐

Then,

N(odd)= 1+4 = 2+3 = 5

Now , we have 1,2,3,4,5. Then

Lemma A-6:(The Birth of Six) 1,2,3,4,5,N

𝑮 = 𝟏, 𝟐 𝑴 = 𝟑 𝑭 = 𝟒, 𝟓

N(even)= 𝐺1+𝐹1 = 𝐺2 +𝐹2= 2× M

Then,

N(even)= 1+5 = 2+4 = 2× 3 =6

Now , we have 1,2,3,4,5,6. Then

Lemma A-7:(The Birth of Seven) 1,2,3,4,5,6,N

𝑮 = 𝟏, 𝟐, 𝟑 𝑴 = 𝑵𝒖𝒍𝒍 𝑭 = 𝟒, 𝟓, 𝟔

N(odd)= 𝐺1+𝐹1 = 𝐺2+𝐹2 = 𝐺3+𝐹3

Then,

N(odd)= 1+6 = 2+5 = 3+4 =7

Now , we have 1,2,3,4,5,6,7. Then

Lemma A-8:(The Birth of Eight) 1,2,3,4,5,6,7,N

𝑮 = 𝟏, 𝟐, 𝟑 𝑴 = 𝟒 𝑭 = 𝟓, 𝟔, 𝟕

N(even)= 𝐺1+𝐹1 = 𝐺2+𝐹2 = 𝐺3+𝐹3 = 2×M

Then,

N(even)= 1+7 = 2+6 = 3+5 = 2× 4 =8

Now , we have 1,2,3,4,5,6,7,8. Then

Lemma A-9:(The Birth of Nine) 1,2,3,4,5,6,7,8,N

𝑮 = 𝟏, 𝟐, 𝟑, 𝟒 𝑴 = 𝑵𝒖𝒍𝒍 𝑭 = 𝟓, 𝟔, 𝟕, 𝟖

N(odd)= 𝐺1+𝐹1 = 𝐺2 + 𝐹2 = 𝐺3+𝐹3 = 𝐺4+𝐹4

Then,

N(odd)=1+8 = 2+7 = 3+6 = 4+5 = 9

Now, we have 1,2,3,4,5,6,7,8,9

Lemma A-10:(The Birth of Ten) 1,2,3,4,5,6,7,8,9,N

𝑮 = 𝟏, 𝟐, 𝟑, 𝟒 𝑴 = 𝟓 𝑭 = 𝟔, 𝟕, 𝟖, 𝟗

N(even)= 𝐺1 + 𝐹1 = 𝐺2+𝐹2 = 𝐺3+𝐹3 = 𝐺4+𝐹4= 2×M

Then,

N(even)= 1+9 = 2+8 = 3+7 = 4+6 = 2× 5 = 10

Now , we have 1,2,3,4,5,6,7,8,9,10. And so on.........

When, there is birth of new Number( Newborn), then all numbers(Ancestors) below, share its

magnitude by this Law. For example if 10 is newborn then 9 and 1 both share 10 additively

and give their consideration to 10. In similar way 8 and 2, 7 and 3, 6 and 4, do, and 5 give its

consideration to 10 as 2 times.

A-(2) Evolution of Natural Line: In the previous section, we saw the "situation of birth" of

Natural Numbers. We saw that, how "Ancestors" get align symmetrically{about middle

position(M)} into two group(G,F) and form additively "Newborn"(N). But in this section we

will see, how "Number Line" evolve.

But before going any further details, let look upon "Natural Line"-

1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,,,,,,,,,,,,,,,,,,,,so on........

Evolution of Odd's : Now as we can see that if we do "Successive Addition" on any "Two

Neighbours(Numbers)", then result is always "Odd", as

1+2=3 2+3=5 3+4=7 4+5=9 5+6=11 6+7=13 7+8=15 and so on. If we

denote "Two Neighbours" by 𝑛 𝑎𝑛𝑑 (𝑛 + 1), and (2𝑛 + 1) is symbol for Odd's. Then,

𝑂𝑑𝑑(2𝑛 + 1) = 𝑛 + (𝑛 + 1) ................................(0)

Evolution of Evens' : If we "Twice" any "Single number" of "Number Line" then we

always get an "Even". Even is very intuitive and we do not have to think rigorously for

"Evens'". By definition of even,

𝐸𝑣𝑒𝑛 = 2𝑛 ...............................................................(0)

Now we call this(for even) operation ," Successive Twice" on Ancestor(M), for any even on

Number Line. Now we establish an "Sum" To N, on Numbers, to understand the "evolution"

of Number line.

Sum Up to N: If we see the Number Line then it is clear that the sequence is like-

1,2,3,4,5,6,7,8,9,10,11,12,13,,,,,,,,,,,,,,,,,,,,,N

1,even,odd,even,odd,even,odd,even,odd,even,odd,,,,,,,,,,,,,,,,,,,,,,N

Then

1 + ∑ {2𝑛 + (2𝑛 + 1)} = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11+, , , , , , , , , , , , , , , 𝑁𝑛1

Or

∑𝑁 = 1+ 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11+, , , , , , , , , , , 𝑁

𝑛

1

To understand the algorithm, let see some Lemma's below-

Lemma 1: after 1, the Newborn is twice of ancestor 1, then 𝑛 = 1

𝐹2𝑛 = 2𝑛,

𝐹2 = 2 × 1 = 2(even)

we have 1,2 now Newborn is sum of 1 and 2

𝑂𝑑𝑑(2𝑛 + 1) = 𝑛 + (𝑛 + 1), 𝑎𝑛𝑑 𝑛 = 1, 𝑡ℎ𝑒𝑛

𝐹2𝑛+1 = 𝑛 + (𝑛 + 1)

𝐹3 = 1 + (1 + 1) = 3

Now we have 1,2,3

Lemma 2: 𝑛 = 2, then

𝐹4 = 2 × 2 = 4

𝐹5 = 2 + (2 + 1) = 5

Now we have 1,2,3,4,5

Lemma 3: 𝑛 = 3, then

𝐹6 = 2 × 3 = 6

𝐹7 = 3 + (3 + 1) = 7

Now we have 1,2,3,4,5,6,7

Lemma 4: 𝑛 = 4, then

𝐹8 = 2 × 4 = 8

𝐹9 = 4 + (4 + 1) = 9

Now we have 1,2,3,4,5,6,7,8,9

Lemma 5: 𝑛 = 5, then

𝐹10 = 2 × 5 = 10

𝐹11 = 5 + (5 + 1) = 11

Then we have 1,2,3,4,5,6,7,8,9,10,11,,,,,,,,and so on.

(A-3) Sum of Natural Series up to 𝒏𝒕𝒉 number:

Let we have a series up to 𝑛

1,2,3,4,5, , , , , , , , , , , , , , , , , , , , , , , (𝑛 − 4), (𝑛 − 3), (𝑛 − 2), (𝑛 − 1), 𝑛

then

∑ 𝑛 = 1 + 2 + 3 + 4 + 5+, , , , , , , , +(𝑛 − 4) + (𝑛 − 3) + (𝑛 − 2) + (𝑛 − 1) + 𝑛𝑛1

Now let reverse the element of this series as

∑ 𝑛𝑛1 = 𝑛 + (𝑛 − 1) + (𝑛 − 2) + (𝑛 − 3) + (𝑛 − 4)+, , , , , , , , , +5 + 4 + 3 + 2 + 1

Now as we can see that both sum are equal, we just replace the elements such that first

become last, second become second last and third become third of last. All numbers is

replaced by this algorithm.

So let add one to one both series,

∑ 𝑛𝑛1 = 1 + 2 + 3 + 4 +, , , , , , , , , , , , , , , , +(𝑛 − 4) + (𝑛 − 3) + (𝑛 − 2) + (𝑛 − 1) + 𝑛

∑ 𝑛 = 𝑛 + (𝑛 − 1) + (𝑛 − 2) + (𝑛 − 3)+, , , , , , , , , , , , , , + 5 + 4 + 3 + 2 + 1𝑛1

2× ∑ 𝑛 = (𝑛 + 1) + (𝑛 + 1) + (𝑛 + 1) + (𝑛 + 1)+, , , , , , , , +(𝑛 + 1) + (𝑛 + 1) + (𝑛 + 1)𝑛1

We can see that after addition of both series, we get 𝑛 terms of (𝑛 + 1) 𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒. Then

2 ×∑𝑛 = 𝑛 × (𝑛 + 1)

𝑛

1

∑𝑛 =𝑛(𝑛 + 1)

2

𝑛

1

............................................ Equation(A-3)

(B) Addition verse Multiplication to Trinity (1,2,3): Consider the following algebraic

equation

𝑋 + 𝑌 + 𝑍 = 𝑋 × 𝑌 × 𝑍

Only solution is

1 + 2 + 3 = 1 × 2 × 3 ...................Equation(B)

Hence, (𝑋, 𝑌, 𝑍) = (1,2,3).

"....For One, Two,, Three,,, addition is equal to multiplication or both are constant".

(C) Natural series and Fibonacci series[6] : Here, I suggest readers to look upon the natural

whole number series and famous series by Fibonacci,

1,2,3,4, 5, 6, 7, 8 , 9 ,10,,,,,,,,,,,,,,,,,,Natural series ( origin=0, Root=1)

1,1,2,3,5,8,13,21,34,55,89,,,,,,,,,,,,,,,,, Fibonacci series( origin=0, Root= 1)

As we can see that Lemma 1,2,3 are same for both series. And Origin/root are common

also, but after three lemma, each has different rule for Newborn.

In Fibonacci Series, rule for Newborn is,

Newborn= Sum of First and second Ancestor

𝑁 = 𝐹1 + 𝐹2

or

𝐹𝑛 = 𝐹𝑛−1 + 𝐹𝑛−2 ...........................Equation (C)

𝐹1 = 𝐺1 = 1

Where 𝐹𝑛 is Fibonacci Newborn, 𝐹𝑛−1is First Ancestor and 𝐹𝑛−2 is Second Ancestor. And

𝐹1 = First Fibonacci Number,

Here, it is very interesting that when we see birth of second Fibonacci Number

𝐹2 = 𝐹1 + 𝐹1

As first three Lemma are same for both series then I personally write Fibonacci Series with

single Great Ancestor -

1,2,3,5,8,13,21,34,55,89,144,,,,,,,,,,,,,,,,,,,,,,,,,,,

1+1, Is a Lemma, for finding the second Fibonacci Number, so it must not be consider as

element of Fibonacci series. But if some wise Mathematicians like (1,1) two ones then it is

OK.

Fibonacci Numbers appear as

Root(1),Even(2),Odd(3),Odd(5),Even(8),Odd(13),Odd(21),Even(34),Odd(55),Odd(89),,,,,,

As we can see that in this progression all number align into -E,O,O-E,O,O- format.

So E,O,O- is the unit link of this series. Then it is clear that in this series we will get more

Primes (in a sample of 𝑛 terms) as compare to Natural series. Because there are two odds and

one even in a unit/link of symmetry.

(D) Asmaka-Tantra ( The Number Theory of Aryabhatta)

Meaning of "Asmaka" is "stone" and "Tantra" is for "System/Theory".

(D-1): The Aryabhatta Series

The origin of the modern decimal-based place value notation can be traced to the Aryabhatta

in his text Aryabhatiya (c. 500), he states

In the second verse of Section-"Ganitam" -

Means that " there is a series of

1,10,100,1000,10000,100000,100000,1000000,................from place to place each is ten

times the preceding."

Aryabhatta series can be written as power of ten as

100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 1010, , , , , , , ,,(origin=0, Root=1).

The rule of Newborn is

𝐴𝑛 = 10 × 𝐹1.............................................Equation (D-1)

Where 𝐴𝑛 is Aryabhatta Newborn and 𝐹1is, First Ancestor/ Previous Term, for 𝐴𝑛.

While first Aryabhatta Number, 𝐴1 = 𝐺1=1, this law is very simple and straight

(Like Fibonacci Series ). ,,,,,each Number is ten time of its Ancestor.

In Aryabhatta series, there is no pairing, but First Ancestor Decide the magnitude of

Newborn. and all Aryabhatta Numbers are Even because all are multiple of ten. By this

simple law, we can do verity of mathematical counting. I think it is one of the most beautiful

progression in mathematics.

(D-2): The Great Matrix

Definition D-2:(Primate Position): "The last numeral( the position of unity) in digits is

called(here) Primate, it is the position of, Leader of the digit". like in 54543,Primate of this

digit is 3.

The sieve of Eratosthenes is the beautiful algorithm of finding primes up to any numbers.

But Asmaka-Tantra(Number Theory of Aryabhatta) is another fundamental method for

studying even/odd, Prime and composite. This method is base on "the last position" "The

last Numeral" of digit among all natural whole numbers", called "Akshra" in Sanskrit. As

we know that the on last position 10 possibility of all Numerals.

Second verse of DasaGitika( Stanza of Ten)-

Authors translation:

[ The Square Matrix of Numearls(10× 𝟏𝟎),,, Next matrix,,Next.. Next,,, and so on, From

"Ka"(1) to "Nya(5)" and, Nya(5)+Ma(25)=Ya(30).

0,2,9,1 are Notes(even/odd), then Matrix of Numerals (9× 𝟗)...Next,,,,,Next,,,,Next,,,,and

so on.]

By the word-"vargaakshrarni", this word is sum of two words 1st is "vargA"( Square

Matrix, male/singular or discrimination between two ) and 2nd is "aksharani"( words

of script/notes, here the numerals/plural for Numerals, for Digit, Table/series of

script/notes, ). Aryabhatta used these three word specifically in that way that they can show

dual or triple meaning and these words, having the musical tone with rhythm, are important

for evaluating his work on mathematical music and as a poet. The meaning of this whole

word is "Base Matrix of Numerals or digits" or the square(Base) matrix(10 × 10) 1 to

100. The next word is "varge-varge-vargaakshrani" in this word he used two time

"VargE" with "E" in the last{ while "vargA" is singular and "VargE" is plural} and then

third time he use "Vagaaksraani". This segment means "continuous series of Base

Matrix" (like 1,2,3,,,,and so on). Here must be consider for next consecutive matrix of

(10× 10) 𝑙𝑖𝑘𝑒 1 − 100, 100 − 200,200 − 300, , ,, and so on. The same word show me the

discrimination between even/odd and prime/odd composite.

then he said-" Kaaat Nyamoo Yha" means definitely for "appear as 'Ka to

Nya'(Sanskrit abphabets)".

But " Ka-to-Nya" then it must be noted that "Nya" is 5th from "Ka" in Sanskrit

alphabet sequence. And he state that "Ma" pluse "Nya" equal Ya(25+5=30).

This is definition for writing "Aryabhatta Numbers" by using Sanskrit alphabets.

It is the Natural Matrix( table) of (10 × 𝑁 ), in this matrix we have 10 row and Infinite

column. It contain all natural whole numbers. It is below, I just show the very first 10

member of this table. as we can see that numerals will be at vertical position in column.

Aryabhatta described this The Great Matrix as

In The Grand series of natural numbers this table/matrix, can be seen as the unit matrix/The

Great Matrix(1-100) of that series, as in below table (E-2a)

• 1 11 21 31 41 51 61 71 81 91 ,,,,,,,,,, ,,,so on

• 2 12 22 32 42 52 62 72 82 92 ,,,,,,,, ,,,, so on

• 3 13 23 33 43 53 63 73 83 93 ,,,,,, ,,,,,,,so on

• 4 14 24 34 44 54 64 74 84 94 ,,,, ,,,,,,,,,so on

• 5 15 25 35 45 55 65 75 85 95 ,,, ,,,,,,,,,,,so on

• 6 16 26 36 46 56 66 76 86 96 ,, ,,,,,,,,,,,,so on

• 7 17 27 37 47 57 67 77 87 97 , ,,,,,,,,,,,,,so on

• 8 18 28 38 48 58 68 78 88 98 ,,,,,,,,,,,,,,,so on

• 9 19 29 39 49 59 69 79 89 99 ,,,,,,,,,,,,,,,so on

• 10 20 30 40 50 60 70 80 90 100 ,,,,,,,,,,,,,,so on

Table (D-2a)

Property of this table is that we can see all Primate Number are align in all 10 row.

After stating this series he state in second line of verse, the word" Kh(0)-Dvi(2)-Nav(9)-

Ke(1) Swara". The word "Kh"- is used by Indian mathematicians( Brahamgupta, Bhaskra)

in Sanskrit its mean by Null or represent 0. "Dvi"- means by Two (2) and "Nava"- means

Nine(9), while"-Ke" actually be pronounce ( in Sanskrit) as "Eke"-means (1) , in pairing of

"Nav+Eke=Nav-ke" if pronouns accurately then he say-" 0-2,9-1", while word "Swara"

meaning for Musical notes or Even/odd. After defining even-odd.

He said word "Nav"(9) and then "Varge,,,, Varge" but pronunciation is "

Vargee...Avarge", then he defiantly talking about " squares of 9`s of Even...Odd". But all

even/odd are 10 not 9 ?

If we want do this then we have to delete 1st column and 10th row of Grand Matrix. Then we

find a symmetry about row(5), means for( Square...Square of nine) is also (9× 9 = 81 ) in a

unite of new square Matrix of 9 " nav" order of matrix of 9 order inside, The Grand

Matrix(10× 10).

He then state that-" Nav(9) Varge-varge Nav-a-antya-Varge Va". this means that -"after

doing so we get a square series of 9(32) × 9(32)…9 × 9… .9 × 9… .9 × 9… 𝑠𝑜 𝑜𝑛… ..",

Since he used word"varge-varge" two times to show that nine is also a square of three (32 =9) and discrimination between something (two quantities). The word -"Nav-a-antya-Varge

Va" means -"Endless series of new base matrix of 9× 9 order".

The resulting series is below-

Method: Do not consider row 1 and column 10( in Grand Matrix), then we get a

series(11-99) of " Square of 9`s(1,2,3,4,5,6,7,8,9 primates)" in the Grand Matrix.

• 11 21 31 41 51 61 71 81 91 ,,,,,,,,,,,,,,,so on

• 12 22 32 42 52 62 72 82 92 ,,,,,,,,,,,,,, so on

• 13 23 33 43 53 63 73 83 93 ,,,,,,,,,,,,,,,so on

• 14 24 34 44 54 64 74 84 94 ,,,,,,,,,,,,,,,so on

• 15 25 35 45 55 65 75 85 95 ,,,,,,,,,,,,,,,so on

• 16 26 36 46 56 66 76 86 96 ,,,,,,,,,,,,,,,so on

• 17 27 37 47 57 67 77 87 97 ,,,,,,,,,,,,,,,so on

• 18 28 38 48 58 68 78 88 98 ,,,,,,,,,,,,,,,so on

• 19 29 39 49 59 69 79 89 99 ,,,,,,,,,,,,,,so on

Table (D-2b)

This series contain all naturals, except primates of 0`s and first decade.

We can see the symmetry about Primates Line(5th). It cannot contains primes other

then 5 (first element),because others will be odd multiples of 5.

(D-3) Ten Families of Natural Numbers : On the base of ,last position of digit( the

primate position), of numbers

1. Row First, contain all primates of 1 or all numbers that have 1 in last

position.

2. Row Second, contain all primates of 2.

3. Row Third, contain all primates of 3.

4. Row Fourth, contain all primates of 4.

5. Row Fifth, contain all primates of 5. i.e., all odd multiples of 5.

6. Row Sixth, contain all primates of 6.

7. Row Seventh, contain all primates of 7.

8. Row Eighth, contain all primates of 8.

9. Row Ninth, contain all primates of 9.

10. Row Tenth, contain all primates of 0.i.e., all even multiples of 5.

But as we can see that the row (2) have only prime 2 at first position. And in row (5), 5 is

only Prime( because all other elements are odd multiples of 5 ).

If we see, the definition of primes, then it is clear that primes( greater then 2) must be odd.

But Primates of 5( except 5) cannot be prime( because all will be composite of odd multiples

of 5.

(D-4)The Cast and classification of Natural Numbers on base 2:

The Hindu-Arabic Numerals (0,1,2,3,4,5,6,7,8,9) are the tones{ Aryabhatta said them as

swara means-notes of music} of mathematical music. We can form and write any number.

But my focus, here is on the Last position in natural numbers. It is the deciding factor in the

nature of these numbers. This last position of Numeral, goes to Primate Numeral for any

particular number.

If the last numerals/Primate is one of 0,2,4,6,8. Then, it is confirm that this number will be

Even, otherwise Odd. so for all even numbers digit The Last Position is reserve for multiples

of 2.

Prime numbers, who resist internal pairing(factorization). it is the specific nature of any

prime number.

Composite numbers are opposite to Prime numbers. They can be written in multiples.

Six Cast of Two Nature(even/odd) in three class for all 10 Family, with respect to, prime

factors of numbers

(I)(a) Even, 𝟐𝒌 ( Vargas ) /( 2,4,8,16,32,64,128,256,512,1024,,,,): This cast is belong to

Four primates of 2,8 and 4,6

(b) Multiples of 2,(2𝒌) called (here) Even Composite. Although it contain (odd) in

multiplication but output is even. But we don`t consider those even who are even multiples

of 5 like 10,20,30,40,50,,,,i.e. primates of 0. Primates of this cast are 2,4,6,8.

(II)(a) Odd Numbers(𝟐𝒌 + 𝟏) (Aavargas ), this cast of number is group of Four primates

family of 1,9 and 3,7 without primate 5.

(b) Multiples of only Odd numbers(primes), called Odd Composite.( examples-63 =21 × 3 ), not contain multiples of 2. But odd number those are multiples of 5, are not in this

cast, like 5,15,25,35,45,55,,,,,

(III) Cast of Primate 0,5: Even and Odd multiples of 5, or 5𝑛 is called(here) primates of 0

and 5 respectively.

(a) Odd multiples of 5 is symmetry line. The Primates of 5.

(b) While, even multiples of 5 become Decades. The Primates of 0.

Even (a) 𝟐𝒌 (b) Multiples of 2𝒌

Without even multiples of 5

Odd (a) 𝟐𝒌 + 𝟏, (b) Multiples without 5

Primates of 0,5 (a) Primate of 0 (b) Primates of 5

Table (E-2d)

Then, The Fundamental Theorem of Arithmetic`s can be represented as

𝑁! = 2𝑘 × 5𝑘 .∏ 𝑝𝑖𝑘

𝑘

𝑖=1

where 𝑖=1,3,7,11,,,,,,,𝑝𝑘, i.e. 𝑘 ≠ 2,5.

After this classification, we can see that even/odd(class) have two-two cast( of four-four

Primates). While Primates 5 and Primates of 0 are two different cast and form third class of

5.

"This is Aryabhatta classification".

In this classification numbers have two fundamental nature(Even/Odd), two secondary

nature(Prime/Composite). Then all have 10 families, 6 cast and 3 class.

Explanation: If we have a container full of Natural Whole Numbers. Now, we remove first

of all 2𝑘 . Then we have remove all remaining even 2k But not Primate of 0. Then we have to

remove all composite odd(multiples of Primes) But not Primate of 5. then we have to remove

All Multiples of 5,i.e. Primates of 0,5.

Then now we have in our container all Primes and 1. now if we remove all prime, then we

have single element 1 in this container.

After looking above I have intuition that there is a very week almost zero percent chance

that , there will not be, three prime like (2,3,5) in naturals. This Primes Trinity, is rare to

number line. This is the nearest Primes Trinity (2+3+5=10).

It is clear that row 5 show some symmetry for distribution of Primes. Here, I must state that-

" I have a personal appeal to this line that it is connected to Riemann Zeta Function and the

Critical Line of that function. Because Critical line(1/2) show symmetry(vertical) on Half of

unit. And this line of 5 is also showing symmetry (horizontal) on half of this table unit(10).I

feel that Riemann Hypothesis must be right."

Section-(E) : Symmetrical Digits

Symmetrical Digit(SD) (𝒑𝒏): There are type of digit (>10), like 11,which are called(here)

Symmetrical digit(SD) on Natural Number Line. They can have only one Numeral among

1,2,3,4,5,6,7,8,9 while we indexing them on Number Line. Some examples of SD are

33,44,55,777,99999999...E.T.C.

While symmetry in digit of numeral 0 are 100,20000,5000000 E.T.C. This numeral(0), must

have another numeral to hold the place of digit along with 0.Therefore,

𝑝 = 𝑁𝑢𝑚𝑒𝑟𝑎𝑙𝑠/𝑃𝑟𝑖𝑚𝑎𝑡𝑒𝑠, 𝑛 = 𝑝𝑙𝑎𝑐𝑒 𝑜𝑓 𝑑𝑖𝑔𝑖𝑡

Notation:

If we have digit 2222222222.

Then 𝑝 = 2, 𝑛 = 10. Therefore

2222222222=(𝟐𝟏𝟎)

For SD, 𝑛 ≥ 2 𝑎𝑛𝑑 𝑝 = 0,1,2,3,4,5,6,7,8,9

Theorem (E): Every Symmetrical digit ,which made up by single numeral 0,2,3,4,5,6,7,8,9

greater then 10, will be composite.

𝑝𝑛 = 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒, ∀ 𝑝 = 0,2,3,4,5,6,7,8,9 𝑎𝑛𝑑 𝑝 ≠ 1,𝑛 ≥ 2

Proof: Let there (𝑝𝑛) is any digit made up to 𝑛𝑡ℎ place with numeral/primate 𝑝.

𝑝 = 2,3,4,5,6,7,8,9, and 𝑝 ≠ 1

Then, we have

(𝑝𝑛) = 𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝, , , , , , , , , , , , , 𝑝𝑛

(𝑝𝑛) = 𝑝 × (1111111111111111111, , , , , , , , , , , , , , 1𝑛)

(𝑝𝑛) = 𝑝 × (1𝑛)

Hence proved that (𝑝𝑛) is composite. If 𝑝 = 0,2,3,4,5,6,7,8,9 𝑎𝑛𝑑 𝑝 ≠ 1.

conjecture(E): All digit made up with only numeral 1(except 11), will be composite?

(1𝑛) = 𝑐𝑜𝑚𝑝𝑜𝑠𝑖𝑡𝑒 for all 𝑛 >2.

For examples: 111(37 × 3), 1111(101 × 11), 11111(41 × 271), 111111(37037 × 3).

Hence, all SD( except 11) will be composite?

Section (F) Goldbach Conjecture:

Statement - "Every even number (greater then 2) can be expressed as the sum of two

Primes".

Explaination: If 𝑛 be the even number (large enough), then we have a natural series up to 𝑛,

1,2,3,4,5, , , , , , , , , , , , , , , , , , , , 𝑀, , , , , , , , , , (𝑛 − 5), (𝑛 − 4), (𝑛 − 3), (𝑛 − 2), (𝑛 − 1), 𝑛

Then middle term/Ancestor,𝑀 =𝑛

2 (M can be even or odd)

No. of SYP(𝑠) =𝑛

2

s G F

1 1 (𝒏 − 𝟏)𝒐𝒅𝒅

2 2 (𝑛 − 2)𝑒𝑣𝑒𝑛

3 3 (𝒏 − 𝟑)𝒐𝒅𝒅

4 4 (𝑛 − 4)𝑒𝑣𝑒𝑛

5 5 (𝒏 − 𝟓) 𝒐𝒅𝒅

And so on...to 𝑛

2 And so on...

𝑛

2 And so on...

𝑛

2

Table F

Then ℕ = 𝐺𝑠 + 𝐹𝑠 where 𝑠 = 1,2,3,4,5, , , , , , , , , , , , , , 𝑡𝑜 𝑛

2

As we can see by law of three, at the instance of birth, every Even Newborn is sum of all

SYP (s=1,3,5,7,9,,,,,,,odd pairs and s=2,4,6,8,10,,,,even pairs) and twice of M.

𝑁 = {

𝑠𝑢𝑚 𝑜𝑓 𝐸𝑣𝑒𝑛 𝑆𝑌𝑃 𝑠𝑢𝑚 𝑜𝑓 𝑂𝑑𝑑 𝑆𝑌𝑃

𝑡𝑤𝑖𝑐𝑒 𝑜𝑓 𝑀𝑖𝑑𝑑𝑙𝑒 𝐴𝑛𝑐𝑒𝑠𝑡𝑜𝑟

OR

𝑁 = {∑𝑃𝐸∑𝑃𝑜2 × 𝑀

If we consider only odd pairs for Goldbach Conjecture. Then Law of Three confirm that-

(I)"Every Even Newborn, will be sum of two Odd Ancestors". Above is the algorithm for

being odd( and as well as prime>2) symmetrical pairs form even number additively.

since all prime greater than 2 are odd then it confirm the conjecture.

(II) if we reverse the statement of that conjecture vice versa then it become-"Every two

Primes(>2) will form even number additively". it is a self proved statement.

Then Theorem of Trinity confirm that: "Every even Number(>2) will be sum of SYP of

Primes".

But as we know "The Number Theory of Aryabhatta" in Eq. (D-4)(IV)

𝑜! =𝑁!

2𝑛 × 5𝑛

this statement confirm that every odd number will be either Prime(>2) or multiple of

odd(Prime>2) numbers. and all Primate of 5 will be composite.

then in place of Odd number, we have two possibility.

(a) Prime odd ( PO)

(b) composite odd (CO)

But in Odd SYP , we will get Three type of Pairs( Three Possibility)

(1) CO-CO (2) CO-PO (3) PO-PO

The 3rd is desired condition for Goldbach Problem,

Now we expend the conjecture to Least condition,

The choice of Number Line to chose among Prime and composite: We have only tool to

understand the Goldbach condition of Primes pair, is the choice of number line( when

numbers take position on Number line). As we see above that on number line two alternate

even-odd position occur. And for every odd position there are two choice, first is Prime

odd(PO) and second is Composite odd(CO). we will investigate The Choice of number line

on the occasion of birth of first few even numbers. Because, The rule of formation is valid

throughout Number line and we only get verification by observing initial segment of Number

line. Then,

As in

lemma A-2, F=G so N=1+1 is only symmetrical pair(SYP) for birth of 2. 1 is not composite

nor prime or even/odd. It cannot be relate to these definitions. it is neutral.

Lemma A-4, G=1,M=2,F=3 the one even SYP 2+2 and odd SYP is 1+3 in this There is

First odd and prime number 3 appear at Goldbach Location( all odd pairs O-O ) for even

numbers.

Lemma A-6, G=1,2 M=3 F=4,5

SYP are 1+5 2+4 3+3

there are one even SYP 2+6 and; two odd SYP are 1+5 and 3+3 All element are primes

and we can see the first Goldbach Prime (3+3).

Lemma A-8, G=1,2,3 M=4 F=5,6,7

SYP pairs are 1+7 2+6 3+5 4+4

There is two even SYP 2+6 and 4+4

There is two odd SYP 1+7 and 3+5

all members of odd pairs is prime and 3+5 is Goldbach primes.

Lemma A-10, G= 1,2,3,4 M= 5 F= 6,7,8,9

There are five SYP; 1+9 2+8 3+7 4+6 5+5

Two out of five are even SYP; 2+8 4+6

Three out of five are odd pairs; 1+9 3+7 5+5

3+7 and 5+5 are Goldbach primes

As we can see that 9 is the first odd composite who is appear at the birth on 10.

It can be seen easily that the propagation of being prime in desire odd symmetrical position/

Goldbach position is increasing as even magnitude increase. for large enough even magnitude

we will get more Goldbach prime SYP.

As we can see the birth of first five(2,4,6,8,10) and after this we can say in this algorithm,

there is priority for primes against the odd composite or even composite as 4( get birth after

two prime 2,3) is first even composite and 9( get birth after two more primes 5,7) is first odd

composite. Then Number line make self Choice among Prime/composite.

This theorem show us that -"The symmetrical position(of odd pairs), fill on priority for

Primes, then for odd composite numbers".

Primes are building blocks of Number Line, while Composites are the filling material for

the gaps created by this law for formation of Natural Number line.

But this is not sufficient to prove this conjecture. But we got a proper statement

equivalent to Riemann Hypothesis:

"All even numbers (>2) can be written as sum of symmetrical primes".

Since all Primes are get distributed symmetrically about Middle Position ( 𝑵

𝟐 ) to form

additively that even N.

Referances"

[1]Clark, Walter Eugene; "The Ariyabhatiya of Aryabhata"; The University of Chicago Press,

Chicago,Illinois,1930

[2] Shukla, Kirpashankar and Sharma, K.V; "Aryabhatiya of Aryabhata; Indian National Scince

Academy, New Delhi, 1976

[3] Ayyangar, A.A. Krishinaswami; "The Mathematics of Aryabhata"( A paper read before , The Myhic

Socity)

[4] Deviedin, Mahamahopadhyaya Sudhakara; "BrahmasphutaSidhanta ", The Medical Hall Press,

Benares ,1902

[5] Sharma, Ram Swarupa; "BrahmaSputhaSidhanta", Indian Institute of Astronomical and Sanskrit

Research, New Delhi,1966

[6] Falcon, Sergio." Generalized (k, r)–Fibonacci Numbers", Gen. Math. Notes, Vol. 25, No. 2,

December 2014, pp.148-158 ISSN 2219-7184; Copyright c ICSRS Publication, 2014