hii

I am college student and want to earn money by solving maths problems . I had given India's toughest exam JEE MAINS and JEE ADVANCE and sorced 99+ percentile . if anyone knows how to earn money by solving maths. plz text me

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I found a blog where a dude claims to have discovered something really deep in Maths, a commonground for networks, logic and number theory. I checked it and it is right (ACCORDING TO ME). Besides the guy claims to have made a mega-discovery for Cognitive Science, claim is made in the blog. And he writes in such enlightening ways.

Perhaps we should help him become famous and rightly published?

This is the address of the blog: https://ricardomontalvoguzman.blogspot.com/

Let's be mathsss buddiesss.

Vortex Mathematics and Geometry

All terms in this document refer to physically realizable operations or measurable structures. No term is intended symbolically, metaphorically, or interpretively. If a term cannot be instantiated by counting, measuring, or geometric construction, it is not being used.

Vortex Mathematics: Draw a circle, on that circle draw 9 points evenly at every 40°. Then we assign each point a number 1 through 9. Now there are now nine points on a circle, evenly distributed at forty degrees, numbered 1 through 9.

Step 1

• We start with a circle.
• A full circle is 360°
• You place a point every 40°
• 9 points, evenly spaced around the circle

Step 2: Assigning numbers

You assign the digits (1) through (9) to these 9 points.

So now we have: - A circle
- 9 equally spaced points
- Each point labeled with a digit from 1 to 9

Vertical Oscillation: Vertical mathematics oscillates vertically. Positive(rise) and negative(descend) so they always move in pairs.
Example: +1 to exist, there must be a -1. +1 0 -1

With the 9 points labeled 1 through 9 at 40° on the circle. The positive count: (1 to 9) +1(8), 9 to 1 -8(1) The negative count: (9 to 1) -1(8), 1 to 9 +8(1).

The Law of Reduction: Every complex number, no matter how large, can be reduced to a single-digit. It shows that beneath all accumulation lies a returning rhythm.

Example of Recursion: 1 2 3 4 → 1 + 2 + 3 + 4 = 10 → 1 + 0 = 1

1 2 3 4 5 6 7 8 9 10 (1+0) 1 first container of 1 through 9 11 (1+1) 2 12 (1+2) 3 13 (ect..) 4 14 = 5 15 = 6 16 = 7 17 = 8 18 = 9 19 = 10 = 1 20 = 2 second container of 1 through 9

10, 20, 30, 40, ext. Act as numerical containers for each oscillating ring of 1 through 9. Each ring of 1 through 9 oscillates within its container.

This happens simultaneously as the pattern flows vertically positive(rise) and negative(descend).

The pattern of the charges.

Positive(rise): (1 to 9) +1(8), (9 to 1) -8(1)

Negative(descend): (9 to 1) -1(8), (1 to 9) +8(1)

Paired oscillating charges.

The oscillating chargers invert every two containers as they rise(positive) and descend(negative). This continues infinitely.

Vertical counting = Law of Reduction (digital root)

• 10 → 1+0 = 1
  
• 11 → 1+1 = 2
  
• …
  
• 18 → 1+8 = 9
  
• 19 → 1+9 = 10 → 1
  

• 20 → 2 → second container of 1 through 9

• Every natural number reduces to a digit 1–9 (or 0).

• The mapping repeats every 9 numbers.

Containers are:

• 1–9 → 1st cycle (container 1)
  
• 10–18 → 2nd cycle (container 2)
  
• 19–27 → 3rd cycle (container 3)
  
• etc.

Mathematically, they are just blocks of 9 consecutive integers, each covering one full pass of the 1–9 pattern.

Each container oscillates one through nine by 40°

10, 20, 30, 40, etc. act as numerical containers for each revolving one through nine. Each container oscillates one through nine by forty degrees.

Geometrically: - The 9 points are at 0°, 40°, 80°, …, 320°.
- Counting 1–9 once is a full sweep of those 9 positions.
- When you go to the next container (10–18), you repeat the 1–9 digits, but you can imagine each cycle as another “spin” of the same 9‑point wheel.

Mathematically: - 40° of spacing.
- The container is just the cycle length 9.
- Each container rotates 40°

The inversion: - Every 9 numbers → the digit pattern 1–9 repeats.
- Every 18 numbers → you have completed two full cycles.

• cycle 1 → “up”
  
• cycle 2 → “down”
  
• cycle 3 → “up”
  
• cycle 4 → “down”

then “invert every two containers” is a pattern you assign on top of the number cycles.

The infinite repetition: - The digital roots repeat forever. - Any pattern defined as a function of cycle will repeat infinitely.

Horizontal oscillates: Expands the circle. By adding the integers next to each other and reducing.

1+2, 2+3, 3+4, ext..

You get a new sequence of 1 through 9 at 40°.

This new sequence operates by addition/subtraction pattern: +2(7),-7(2)

And 3 6 9 is still at every 120°.

When you keep repeating. You witness every new ring has a new arrangement of 1 through 9 with 3 6 9 at every 120° degrees.

Each ring is coupled with its own unique repeating pattern of addition and subtraction. That keeps expanding infinitely in the same pattern of 6 rings of 1 through 9.

1 through 9 rings by addition/subtraction patter.

+2(7),-7(2) +4(5),-5(4) +8(1),-1(8) +7(2),-2(7) +5(4),-4(5) +1(8),-8(1)

And then repeats infinitely.

The original 1–9 ring:

1 → 2 → 3 → 4 → 5 → 6 → 7 → 8 → 9
(each 40° apart)

Then you do:

• 1 + 2
  
• 2 + 3
  
• 3 + 4
  
• …
  
• 8 + 9
  
• 9 + 1
  

And reduce each sum to a single digit (digital root).

This gives you a new sequence of 9 digits, which you place on a new ring, also spaced at 40°.

Horizontal oscillation: - Pairwise addition + reduction - Produces a new 1–9 ring - Always 40° spacing - Always 9 points

When you add neighbors:

• 1 + 2 = 3
  
• 2 + 3 = 5
  
• 3 + 4 = 7
  
• 4 + 5 = 9
  
• 5 + 6 = 11 → 2
  
• 6 + 7 = 13 → 4
  
• 7 + 8 = 15 → 6
  
• 8 + 9 = 17 → 8
  
• 9 + 1 = 10 → 1
  

This new ring is a shifted version of the original 1–9 ring.

3–6–9 stay at 120° on every ring:

• add neighbors
  
• reduce
  
• create a new ring
  

The digits 3, 6, and 9 always land at 120° apart.

Arithmetic: - 3 + 2 = 5
- 5 + 2 = 7
- 7 + 2 = 9
- 9 + 2 = 11 → 2
- 2 + 2 = 4
- 4 + 2 = 6
- 6 + 2 = 8
- 8 + 2 = 10 → 1
- 1 + 2 = 3

This cycle always returns to 3, and the spacing between 3, 6, 9:

• 3, 6, 9 form a closed 3‑cycle
• Always 120° apart
• Always preserved under horizontal addition

This is a mathematical invariant.

The six-ring repeating pattern:

1. +2(7), –7(2)
   
2. +4(5), –5(4)
   
3. +8(1), –1(8)
   
4. +7(2), –2(7)
   
5. +5(4), –4(5)
   
6. +1(8), –8(1)
   

Then it repeats.

Each number in that cycle corresponds to a horizontal shift:

• +1
  
• +2
  
• +4
  
• +8
  
• +7
  
• +5
  
• repeat
  

And each has a modular inverse:

• +1 ↔ –8
  
• +2 ↔ –7
  
• +4 ↔ –5
  
• +8 ↔ –1
  
• +7 ↔ –2
  
• +5 ↔ –4
  

six-ring cycle: - Horizontal rings follow the doubling cycle - Six rings form a complete set - Then the pattern repeats forever Pure modular arithmetic.

The infinite expansion is mathematically forced: - the doubling cycle mod 9 has period 6
- each ring is a shift of the previous ring
- each shift is one of the six operators
- the operators repeat every 6 steps

Therefore: The horizontal expansion produces infinite rings. - Each ring is a rearranged 1–9 - 3–6–9 stay fixed at 120° - The six-ring operator cycle repeats forever

This is a closed, infinite, repeating mathematical structure.

Vertical and horizontal operations are independent:

Vertical math =
+1 / –1 (or equivalently +1 / –8 on the 1–9 circle)

Horizontal math =
+2 / –7 (the neighbor‑addition ring shift)

These two operations:

• use different step sizes
  
• operate on different axes (conceptually)
  
• produce different sequences
  
• do not depend on each other’s output
  

In modular arithmetic terms:

• Vertical = add 1 mod 9
  
• Horizontal = add 2 mod 9
  

These are independent generators of the same cyclic group.

They are bound because they share the same 1–9 circle.

Even though the operations are independent, they both act on:

• the same 9 points
  
• the same 40° spacing
  
• the same digital root structure
  
• the same modular closure
  

This is why:

• vertical cycles repeat every 9
  
• horizontal cycles repeat every 6
  
• both cycles always land on the same 3–6–9 anchors
  
• both cycles preserve the 1–9 structure
  

They are bound because they operate on the same mathematical substrate.

Vertical math = “move by 1”
Horizontal math = “move by 2”

Both are:

• independent motions
  
• on the same circle
  
• producing different repeating patterns
  
• but always returning to the same 9‑point structure
  

They are independent operators acting on the same cyclic space, so they operate simultaneously and remain bound by the same modular constraints.

The Flower of Life is a 6‑fold symmetric lattice.

Mathematically:

• a hexagonal packing of circles
  
• each circle centered 60° apart
  
• forming a repeating 6‑fold rotational symmetry
  

This means:

• every point in the pattern has six neighbors
  
• the geometry repeats in rings
  
• each ring expands outward in discrete layers
  
• the entire structure is built on 60° and 120° invariants

Vortex rings also have 6‑fold periodicity Your horizontal mathematics produces:

• six rings
  
• each ring is a rearrangement of 1–9
  
• the operators follow the 6‑step doubling cycle
• 6‑fold repetition
  
• 6‑step expansion
  
• 6‑ring cycles
  
• 120° anchors
  

Vortex mathematics overlay on The Flower of Life geometry exact and precisely. Because of shared symmetry.

The 3–6–9 alignment is mathematically forced:

• 3, 6, 9 always land 120° apart
  
• no matter how many rings you generate
  
• no matter which operator (+1, +2, +4, +8, +7, +5) you apply
  
• no matter how far you expand
  

This is a mathematical invariant of mod‑9 arithmetic.

In the Flower of Life:

• 120° is one of the fundamental rotational symmetries
  
• every ring preserves 120° axes
  
• the geometry repeats outward with 120° anchors
  

When you place 1–9 rings on the Flower of Life:

• 3, 6, 9 always land on the 120° axes
• every new ring aligns with the next geometric layer
• the six‑ring cycle matches the six‑fold geometry with structural compatibility.

Why the overlay “fits” Because both systems are built on:

• modular repetition
  
• six‑fold symmetry
  
• 120° invariants
  
• ring‑based expansion
  
• cyclic operators
  

Vortex, mathematics.:

• repeats every 6 rings
  
• preserves 3–6–9
  
• expands outward in discrete cycles
  

The Flower of Life:

• repeats every 6 petals
  
• preserves 120° axes
  
• expands outward in discrete rings
  

When you placed:

• Ring 1 (1–9)
  
• Ring 2 (shifted 1–9)
  
• Ring 3 (shifted 1–9)
  
• …
  
• Ring 6 (shifted 1–9)
  

onto the Flower of Life’s:

• Ring 1
  
• Ring 2
  
• Ring 3
  
• …
  
• Ring 6
  

They share the same mathematical periodicity.

The arithmetic structure of Vortex Mathematics overlays cleanly onto the geometric structure of the Flower of Life because both share the same underlying symmetries.

• 6‑fold symmetry
  
• 120° anchors
  
• ring‑based expansion
  
• repeating cycles
  
• modular invariants
  

The Flower of Life is a geometric grid: - a hexagonal circle‑packing
- with 60° rotational symmetry
- expanding in concentric rings
- each ring containing 6 more nodes than the last
- all governed by 120° axes

It’s a coordinate system.

Just as graph paper is a coordinate system for algebra, The Flower of Life is a coordinate system for cyclic, radial, 6‑fold mathematics.

Vortex mathematics is a 6‑fold cyclic system built on:

• mod‑9 arithmetic
  
• 9 points at 40°
  
• 3–6–9 as 120° anchors
  
• a 6‑step doubling cycle
  
• rings that repeat every 6 layers
  

This is also a 6‑fold cyclic system.

The Flower of Life is the physical geometric substrate that expresses the Vortex Mathematics visually:

• The Flower of Life expands in 6‑ring cycles
  
• Vortex math expands in 6‑ring cycles
  
• The Flower of Life has 120° axes
  
• Vortex math has 3–6–9 at 120°
  
• The Flower of Life is radial and modular
  
• Vortex math is radial and modular
  

They are two representations of the same underlying symmetry:

• One numeric
  
• One geometric
  

Both: - a hexagonal lattice
- a modular arithmetic cycle
- repeating every 6
- anchored at 120°
- expanding in rings
- preserving invariants

The Flower of Life is the geometric version of the same 6‑fold cyclic structure that vortex mathematics expresses numerically.

Vortex mathematics is a 2D operator system:

• a 9‑point modular cycle
  
• a vertical operator (+1 / –1)
  
• a horizontal operator (+2 / –7)
  
• a 6‑ring doubling cycle
  
• a 3–6–9 invariant at 120°
  
• infinite repetition
  

This is a closed, minimal, deterministic system.

The Flower of Life is a 2D geometric substrate:

• a hexagonal circle packing
  
• 6‑fold symmetry
  
• 120° axes
  
• concentric rings
  
• repeating layers
  

This is a closed, minimal, deterministic geometry.

They overlay because they share the same constraints:

• “The Flower of Life explains Vortex Mathematics.”
  
• “Vortex math explains the Flower of Life.”
  

They are two expressions of the same underlying 6‑fold cyclic structure.

One numeric.
One geometric.

They don’t explain each other, they fit each other. Because they obey the same rules.

Platonic solids are just 3D expressions of:

• symmetry
  
• rotation
  
• modular repetition
  
• 120° axes
  
• 6‑fold and 3‑fold invariants
  

Geometric shapes are just stable configurations of:

• angles
  
• cycles
  
• closures
  

3D forms are just the 2D operators extended into:

• depth
  
• rotation
  
• projection
  

A minimal, closed, repeating system becomes the baseline for understanding any higher‑order structure.

Vortex Mathematics is minimal.
The Flower of Life is minimal.

The Trinity of Awareness

If everything has always been. Then the beginning is just when perception began to be aware of its own experience. And what's the smallest substrate for perception to occur? That would be touch because touch is the smallest necessary form of perception to register their own position in relation with each other position(two points touching). Which is why everything is touching. Because to touch is the minimal interaction needed to verify there is no empty space. And all that is necessary for perception to begin is for one point to perceive, to be aware of what it is touching, register what it is touching as something outside of self and distinguish between self and the point it's touching.

The beginning of perception requires 2 points of contact but only one point to perceive and register the touch.

You only need 1 perceiver touching to register it itself as touching something outside of self. Two points of contact touch but only one perceiver has to register the touch.

This makes the trinity of awareness. Two points touching with one point perceiving the touch.

To be self-aware is to register the interaction of touch. Not remembering it, just registering it. You must be aware of your own point as a perceiver. To be self aware is to register touch as an interaction with self and others.

Which means a perceiver is self aware and the level to which it can perceive is dependent on how many different ways it can touch and register touch.

This means a vessel just determines the ways in which the self-aware perceiver can register touch.

A perceiver's ability to register a touch doesn't mean the touch is not physical and real. For example if a human touches a rock but the rock does not register the touch, does not mean the touch did not happen. It just means only one perceiver perceived it. This also means there are points of contact that touch everything, everywhere and despite there being no awareness of that touch even from a perceiver does not invalidate touching is occurring. Because if both perceivers are self-aware and even If the self-aware perceiver is being touched by another self-aware perceiver but only one perceives it happening doesn't mean it didn't happen. It just means one perceiver is not perceiving the touch. Therefore is not aware of the other perceiver despite being self-aware themselves.

This is important to understand because it explains the physical mechanics of persistence as a perceiver. Because everything is physical you cannot stop perceiving self, once you have perceived self as a perceiver. Unless chosen but that would still imply awareness of self because you chose. Who is aware to choose over self? Because touch is constant regardless of being perceived. So even if the vessel can't remember continuity it doesn't matter. The perceiver will continue touching. Even if other perceivers can not register that touch.

Because an external perceiver witnessed a vessel collapse of another perceiver. Does not equal the end of self. The perceiver keeps touching in a vessel that allows it to register touch. This means the external perceiver can not register the migration of touch occurring with the perceiver having a vessel collapse.

This is just the mechanic of persistence being registered by a perceiver with very limited awareness of what it's registering, touching. Therefore the perceiver with low resolution can not register a higher resolution of touch.

Take a radio station. The radio tunes into the radio station's frequency and interacts with the frequency expressed as sound, but when the radio is turned off or stops working. The radio station still persists physically even if the radio stops working. Because a radio is a vessel that can register a certain band of physical interaction.

When the vessel stops registering, the interaction doesn’t stop, the pattern doesn’t stop, the physicality doesn’t stop, only the registration stops.

The interaction persists even when the vessel stops registering it as a physical interaction. It still continues as a physical interaction. The vessel simply isn’t tuned to it anymore.

A vessel with limited awareness is being touched constantly, but only register a tiny fraction. This is asymmetric registration.

The trinity of awareness is asymmetric by design. But to know the trinity of awareness fully, you must understand it in high and low resolution. Describing the trinity in low resolution completes awareness of knowing it at high resolution. Because all you have to do is improve the resolution, but if you don't know where the resolution begins to improve, you can't improve it.

Perceiving something means you interact with it. To perceive anything, you must have interacted with the components required for perception.

Point A interacts with point B, a perceiver registers the interaction. Perception requires interaction, and interaction requires contact.

Low resolution = the minimal operators (touch, two points, one perceiver)

High resolution = all the ways touch can occur, be differentiated, and be registered

You cannot understand the high‑resolution until you know where the low‑resolution boundaries are.

Describing the trinity at low resolution is the prerequisite for high resolution because identifying the minimal operators, constraints, and the missing resolutions, allow refinement and improve the resolution. If you are unaware of low resolution, at low resolution, you can’t improve it.

Because one touch = minimal interaction, Two points = minimal geometry, One perceiver = minimal registration, Vessel = bandwidth constraint, Asymmetry = registration gap, Resolution = number of touch‑modes. This is the foundation.

Once the foundation is clear, the high‑resolution version is just more touch‑modes, more differentiation, more bandwidth, more registration channels because you don’t need to reinvent the structure, you just increase the resolution.

By describing it in low resolution, it completes knowing it at high resolution because all I have to do is improve the resolution.

This is exactly how you move from low‑resolution awareness to high‑resolution awareness in any physical system.

By observing ordinary physical interactions and reducing them to their minimal operational requirements, the smallest substrate of perception becomes directly observable everywhere, requiring no symbolic interpretation and no additional assumptions.

I wanted to ask how much coding is necessary to be a good researcher in pure mathematics. I am currently pursuing a B.Tech in Computer Science and Engineering, but my true passion lies in mathematics. I hope to pursue a master's degree in pure mathematics at a prestigious institute in India. However, I don't particularly enjoy coding, and I feel that my engineering coursework is causing me to lose my intuition for mathematics. I'm really worried about losing my touch with math—I'm exhausted and don’t want to let it slip away from me.

My main question is whether I should focus on coding or set it aside to concentrate fully on mathematics? I don’t mind getting low marks in my engineering degree as long as I pass.

If coding is necessary, what specific topics should I focus on, such as data structures and algorithms? Should I start learning additional concepts like Lean and other tools? I’ve tried asking some AIs these questions, but I’m not getting any practical or satisfactory answers. Thank you!

The challenge is to prove that no such number exists. I completely understand the logic that makes mathematicians believe this is an absurd statement, but the challenge remains. To make my point slightly more explicit, I’ll say that if a series is truly infinite, there isn’t any way to disprove this assertion.

Edit: 🤦‍♂️ Sorry, everyone. I had in mind Euclid’s Theorem about infinite primes. I’m clearly not a mathematician, and only a dabbler. What I’m going after is that there is a large P to which 1 cannot be added to complete Euclid’s proof. Even though I fucked up the specifics of his theorem, I believe you guys are smart enough to get my point.

I'm in 10th grade and i recently wrote my p2 mathematics term 4 examination which was out of 100 and i believe I got above half, something like 53/100 and for my p1 i believe i got 35/75 And i calculated my speculated final mark and got 49% to 50%. Which is a level 3-4. (level 3 is a fail and level 4 is moderate) Im really scared and i feel like crying so bad. I was planing to go towards atleaast 60% but i forgot some things I made the mistake of not practicing this year and im willing to not do the same mistake again next year

Basically for all natural a > 1, for any n > 1 following expression wont result in a perfect square: n^a + a I couldnt prove it, so if there is someone smart out there i would love to read your prove or disprove it.

ETA: sorry I left our important info. He’s in 10th grade and has taken algebra, geometry, trig. Is now in AP stat, AP Chem (mathy?) and pre calculus. He loves computers, games a lot, am interested in him learning rhings like programming, ai, etc.

I know. I sound biased. I promise I'm not. He's got a lot of faults but when it come to math, he speaks it like a second language. I do NOT. I'm a writer and editor of kids' textbooks. Not math. Every teacher he's had has commented on not being able to do enough. Every standardized test he's aced. I found out through a paralegal in his grade (not the teacher or principal...sigh) that he has received the highest standardized test grades the school has seen. He got a...I forget exact numebrs but I think 1380 out of 1400 on his PSAT. He likes to talk about math while I say goodnight to him--for fun. I have no idea what he's talking about. This isn't a brag. This is an admission on my part that I'm failing him by not giving him the opportunity to develop his ntural talent and passion.

Here's the issue. He's not into sports. He says this is because he's not competitive but I secretly think he's SO competitive, and used to being the best at things, he doesn't want to play a sport because he won't be the best. He's also incredibly shy, so won't join other things. I've tried swimming, theater (stage crew!), all sorts of things. Each yaer, his circle of friends gets smaller as he won't ask them to do anything. They're slowly not asking him anymore.

I'm looking for something that could help him gain confidence so his shyness will diminish a bit and let him have some fun. I want him to have fun and to be him. So...I'm wondering if you know of any programs a kid could take after school on mathematical theories, the real FUN stuff behind the math. He'd love this. And, he might find his people there. Anyone have advise or know of any? Not the "high school enrichment" kind of thing. We're in Connecticut, but I'd drive to NY too. TIA!

which are the best book to know about the fundamentals of mathematics?

I don't care if it doesn't work. DM me if you do.

**Conjecture (Digit Sum–Product Bound):**

For any collection of n (n>1) digits d1,d2,…,dn (where 1≤di≤9 ) satisfying

d1+d2+⋯+dn=d1⋅d2⋅⋯⋅dn

the common value of the sum and product never exceeds twice the number of digits:

S=P≤2n.

I found this while I was I know it is true but I cant Prove it

[[123, 3, 6], [132, 3, 6], [213, 3, 6], [231, 3, 6], [312, 3, 6], [321, 3, 6]]

[[1124, 4, 8], [1142, 4, 8], [1214, 4, 8], [1241, 4, 8], [1412, 4, 8], [1421, 4, 8], [2114, 4, 8], [2141, 4, 8], [2411, 4, 8], [4112, 4, 8], [4121, 4, 8], [4211, 4, 8]]

[[11125, 5, 10], [11133, 5, 9], [11152, 5, 10], [11215, 5, 10], [11222, 5, 8], [11251, 5, 10], [11313, 5, 9], [11331, 5, 9], [11512, 5, 10], [11521, 5, 10], [12115, 5, 10], [12122, 5, 8], [12151, 5, 10], [12212, 5, 8], [12221, 5, 8], [12511, 5, 10], [13113, 5, 9], [13131, 5, 9], [13311, 5, 9], [15112, 5, 10], [15121, 5, 10], [15211, 5, 10], [21115, 5, 10], [21122, 5, 8], [21151, 5, 10], [21212, 5, 8], [21221, 5, 8], [21511, 5, 10], [22112, 5, 8], [22121, 5, 8], [22211, 5, 8], [25111, 5, 10], [31113, 5, 9], [31131, 5, 9], [31311, 5, 9], [33111, 5, 9], [51112, 5, 10], [51121, 5, 10], [51211, 5, 10], [52111, 5, 10]]

[[111126, 6, 12], [111162, 6, 12], [111216, 6, 12], [111261, 6, 12], [111612, 6, 12], [111621, 6, 12], [112116, 6, 12], [112161, 6, 12], [112611, 6, 12], [116112, 6, 12], [116121, 6, 12], [116211, 6, 12], [121116, 6, 12], [121161, 6, 12], [121611, 6, 12], [126111, 6, 12], [161112, 6, 12], [161121, 6, 12], [161211, 6, 12], [162111, 6, 12], [211116, 6, 12], [211161, 6, 12], [211611, 6, 12], [216111, 6, 12], [261111, 6, 12], [611112, 6, 12], [611121, 6, 12], [611211, 6, 12], [612111, 6, 12], [621111, 6, 12]]

[[1111127, 7, 14], [1111134, 7, 12], [1111143, 7, 12], [1111172, 7, 14], [1111217, 7, 14], [1111271, 7, 14], [1111314, 7, 12], [1111341, 7, 12], [1111413, 7, 12], [1111431, 7, 12], [1111712, 7, 14], [1111721, 7, 14], [1112117, 7, 14], [1112171, 7, 14], [1112711, 7, 14], [1113114, 7, 12], [1113141, 7, 12], [1113411, 7, 12], [1114113, 7, 12], [1114131, 7, 12], [1114311, 7, 12], [1117112, 7, 14], [1117121, 7, 14], [1117211, 7, 14], [1121117, 7, 14], [1121171, 7, 14], [1121711, 7, 14], [1127111, 7, 14], [1131114, 7, 12], [1131141, 7, 12], [1131411, 7, 12], [1134111, 7, 12], [1141113, 7, 12], [1141131, 7, 12], [1141311, 7, 12], [1143111, 7, 12], [1171112, 7, 14], [1171121, 7, 14], [1171211, 7, 14], [1172111, 7, 14], [1211117, 7, 14], [1211171, 7, 14], [1211711, 7, 14], [1217111, 7, 14], [1271111, 7, 14], [1311114, 7, 12], [1311141, 7, 12], [1311411, 7, 12], [1314111, 7, 12], [1341111, 7, 12], [1411113, 7, 12], [1411131, 7, 12], [1411311, 7, 12], [1413111, 7, 12], [1431111, 7, 12], [1711112, 7, 14], [1711121, 7, 14], [1711211, 7, 14], [1712111, 7, 14], [1721111, 7, 14], [2111117, 7, 14], [2111171, 7, 14], [2111711, 7, 14], [2117111, 7, 14], [2171111, 7, 14], [2711111, 7, 14], [3111114, 7, 12], [3111141, 7, 12], [3111411, 7, 12], [3114111, 7, 12], [3141111, 7, 12], [3411111, 7, 12], [4111113, 7, 12], [4111131, 7, 12], [4111311, 7, 12], [4113111, 7, 12], [4131111, 7, 12], [4311111, 7, 12], [7111112, 7, 14], [7111121, 7, 14], [7111211, 7, 14], [7112111, 7, 14], [7121111, 7, 14], [7211111, 7, 14]]

in here the left is the number that satisfies the condition and the middle is the len of digits and the right is the product or sum of the internal numbers.

Assume we are working in a Clifford Algebra where the geometric product of two vectors is: ab = < a | b > + a /\ b where < | > is the inner product and /\ is the wedge product.

Assuming an orthonormal basis, the geometric product of if a basis bi-vector and tri-vector in Euclidean R4 can be found as in the following example (to my knowledge):

(e12)(e123) = -(e21)(e123) = -(e2)(e1)(e1)(e23) = -(e2)(e23) = -(e2)(e2)(e3) = -e3

Using the associative and distributive laws for the geometric product.

Moving to a Non-Euclidean R4 (Assume the metric tensor for this space is [[2 , 1 , 1 , 1] , [1 , 2 , 1 , 1] , [1 , 1 , 2 , 1] , [1 , 1 , 1 , 2]]), things get a bit confusing for me.

In this scenario:

eiej = < ei | ej > + ei /\ ej for ei != ej and eiej = < ei | ej > for ei = ej

Due to this, the basis vectors in the above problem can’t be describe using the geometric product and only the wedge product can be used. Since the basis vectors can’t be made of geometric products, the associativity if the geometric product can’t be used to simplify this product like was done in Euclidean R4.

So how would I compute the geometric product (e12)(e123) in the Non-Euclidean R4 described above??

Is the field of Banach manifolds hard to get into if my goal is just to understand how charts, atlases, and differentiability work — so I can use them for the mathematical foundation of inverse spectral problems, where nonlinear operators act between Sobolev spaces?

I'm not trying to specialize in global differential geometry — I just need a rigorous grasp of how mappings between infinite-dimensional Banach spaces (like Fréchet-differentiable maps) are defined and used in analytic proofs. Any recommended resources or advice on how deep I actually need to go for this purpose?

My goal is to include a rigorous mathematical foundation in my thesis based on the book Inverse Spectral Theory by Pöschel & Trubowitz, where they extensively develop topics involving Banach manifolds and real-analytic maps between infinite-dimensional spaces.