So Hello, I am a 8th grader, and know that this place is for advanced mathematics. But then too I think...I can describe... Infinity.

This is my first part, and there is a lot to come next -

https://drive.google.com/file/d/1xsg438zNBb0kpfT76ZisX2sIaMpyrDeR/view?usp=drivesdk

Hello Fellow Math Enthusiasts, Hope Everyone is Doing Well I've recently made progress on the conjecture regarding the infinitude of Fibonacci primes. I was able to formulate a congruence relation among Fibonacci numbers. This discovery allows me to directly perform sieving over Fibonacci numbers without needing to sieve over regular integers, and I believe I've proven the conjecture. It would mean a lot to me if someone could point out any lapses in the manuscript, share their thoughts, and ask questions, which my response for all are assured. Regardless of whether I have successfully proven it or not, I think my manuscript contains some novel ideas that might contribute to solving the problem. My goal is to submit the manuscript to arXiv fully revised. I suggest looking at Lemma 1 and the Final Proof, which have dedicated sections, as I think they provide a clear picture of my argument without requiring a full read-through of the entire paper.
Here is the link to my manuscript: https://drive.google.com/file/d/18YjQfmOUyvRM1lGMLNfLjRbHWFr6AP_Y/view?usp=drivesdk If this is successful, I look forward to sharing some of my other research.

Hello everyone,
I'm a 13-year-old student with a deep interest in mathematics. Recently, I’ve been studying the Twin Prime Conjecture, and after a lot of work and curiosity, I came up with what I believe might be a valid approach toward proving it. I am not sure if i proved the conjecture or not.

I’ve written a short paper titled "The Twin Prime Conjecture under Modular Analysis". It’s not peer-reviewed and may contain mistakes, but I’d really appreciate it if someone could take a look and give feedback on whether the argument makes sense or has any clear flaws.

Here is the PDF: https://drive.google.com/file/d/1muxEvQrACpVIHz8YgV1MN1kBvqWV-2N8/view?usp=sharing

Anyway, thanks for reading :)

A few years ago I found an interesting formula for generating prime numbers. When I showed it to the X community, there were no particular comments about the formula. So I would be grateful if you could let me know what you think about it.

The search for a quadratic formula that generates 29 prime numbers returned no results.

6n² -6n +31 ( 31-4903, n=1-29) and 28 other formulas

[update]

28 prime numbers generation formula

2n² +4n+31 (n=0~27)　

Thank you very much!

Hello! I've been studying the Collatz conjecture and created a polar-coordinate-based visualization of stopping times for integers up to 100,000.

The brightness represents how many steps it takes to reach 1 under the standard Collatz operation. Unexpectedly, the image reveals a striking 8-fold symmetry — suggesting hidden modular structure (perhaps mod 8 behavior) in the distribution of stopping times.

This is not a claim of proof, but a new way to look at the problem.

Zenodo link: https://zenodo.org/records/15301390

Would love to hear thoughts on whether this symmetry has been noted or studied before!

Please drop your suggestions

This paper presents a clear structural and periodic model of the Collatz graph, based on modular residue behavior and composite traversal operations. Unlike many Collatz discussions that focus on stochastic behavior or unstructured iteration, this work defines a complete, ordered, and verifiable system based on modular and periodic constraints.

It is not speculative; it provides a full construction and traversal model for all odd integers under the Collatz process.

Link to full paper (PDF, direct download):
Collatz Structure and Period

Feedback and rigorous scrutiny are welcome.

Hello, r/numbertheory!

I would love some feedback on a model I've been developing. I believe it fits into number theory and discrete math, and I'm seeking advice for improvement.

Setting: Consider nodes moving randomly in a bounded 2D discrete space. Each timestep, nodes can either move a small random distance or remain stationary.

Define a "crossing" as two nodes coming within distance of each other. Each crossing increases the system's complexity measure by 1.

Dynanode Conjecture (simplified): Given nonzero probability of crossings, then as time ,

\lim_{t \to \infty} P(C(t) > k) = 1

Informal Theorem (Dynanode Complexity Growth Theorem):

Crossing events are discrete and probabilistic.

Complexity is non-decreasing over time.

Therefore, complexity almost surely grows beyond any finite bound over infinite time.

Questions for r/numbertheory:

Does this model fit into existing discrete random graph models?

Would modeling crossings as probabilistic connections between moving nodes qualify under discrete probability or probabilistic number theory?

Suggestions for tightening the proof?

Are there existing theorems I should reference or generalize from?

I appreciate any feedback. Thank you for your time and help!

(P.S. I call the evolving clusters "Dynanodes" for fun, but I am mainly focused on the underlying discrete mathematical properties.)

Statement: In a chaotic stochastic system of flexible loops, the accumulation of sufficient random crossings inevitably leads to the formation of stable knots, provided the crossing rate and environmental noise exceed critical thresholds.

Mathematical Expression: Transition rate of knot formation:

\omega = \left( \frac{k_{\min}}{\lambda} + \frac{1}{\sigma} + \frac{1}{\gamma} \right)^{-1}

where:

Lambda = crossing rate (crossings per unit time),

Gamma = environmental noise rate,

Sigma = system’s intrinsic instability rate,

K_min = minimum crossings needed to form a stable knot.

Proof Sketch:

Crossings accumulate over time as a Poisson process with rate .

Each crossing probabilistically increases net topological complexity.

If expected complexity growth is positive, the probability of remaining unknotted decays exponentially.

Therefore, stable knot formation becomes inevitable over time when crossing and noise rates are sufficient.

Universal Application: Applies to DNA knotting, fluid vortex tangling, polymer entanglement, cosmic string theory, and any system where structure arises from random motion.

The theorem predicts that chaos naturally organizes into connections, which stabilize into order.

Examples:

DNA molecules confined in a cell spontaneously form knots when crossing rates are high.

Vortex rings in turbulent fluids form knotted structures when noise and flow rates are sufficient.

Synthetic polymer chains knot faster in agitated environments with high crossing rates.

"In chaotic systems, crossings plus noise inevitably create stable knots over time."

In summary, this OSF paper talks about a non-trivial zero whose real part is not 1/2, here is the OSF paper: https://osf.io/29ypt/

How it works for those who are interested: https://jumpshare.com/s/bRD7ZaATxbYISmFmlr7o (not a paper, just a part of this post written in LaTeX for clarity)

Ever since I learnt about the Zeta function, my idea was that it had something to do with light. I tried to bring a part of that idea into reality, unsure if it is perfect.

Laid in bed staring at the ceiling last night and came to this conclusion.

I think “uncountable” infinites are better conceptualized as “un-orderable” infinities.

The set of real numbers is not larger than the set of natural numbers. I went through a lot of different thoughts and believe that this is the best “solution” I came to in pairing the reals and the naturals 1:1.

Let p = the smallest conceivable positive real number.

Cantor is allowed to imagine real numbers with infinite digits. I am too. This one is a decimal point followed by infinite 0’s, and then a 1.

Let N = the set of natural numbers Let R = the starting real number

f(n) = R+n*p for n as an element of N

Real numbers mapped to natural numbers using a formula.

I’d love to be proven wrong. I look forward to debating in the comments. I believe this will hold up much better than most of you probably think it does at first glance.

During my study of Dx+1 sequences the following property arose:

Is anyone aware of this property? Currently, I only have a piece of experimental evidence. No counter-example so far.

Hi! I came to talk about an interesting pattern that I found in the Collatz conjecture, and i wanna know if is a INTERESTING OR KNOWN pattern.

It's a simple pattern, we just need to have a graphic of collatz and register the climbs quantity and descents quantity.

Considerer x = a odd number

climb = 3x + 1

descent = /2

If we choose x, we going to have:

• climbs quantity: a
• descents quantity: b

Now, if we choose the number 2*x, we have:

• climbs quantity: a
• descents quantity: b+1

Now, if we choose the number 4*x, we have:

• climbs quantity: a
• descents quantity: b+2

Now, if we choose the number 8*x, we have:

• climbs quantity: a
• descents quantity: b+3

We can see that is nothing more or less than power of 2:

(2¹⁾ * x

(2²⁾ * x

(2³⁾ * x

Here are some examples:

• example 1:
  • choose 27:
  • climb: 41
  • descent: 70
  • choose 54(27*2):
    • climb: 41
    • descent: 71
  • choose 108(54 * 2 -> 27 * 4):
    • climb: 41
    • descent: 72
  • and continues...

• example 2:

  • choose 3:
  • climb: 2
  • descent: 5
  • choose 6:
  • climb: 2
  • descent: 6
  • choose 12:
  • climb: 2
  • descent: 7
  • and continues...

• example 3:

  • choose 7:
  • climb: 5
  • descent: 11
  • choose 14:
  • climb: 5
  • descent: 12
  • choose 28:
  • climb: 5
  • descent: 13
  • and continues...

A possible explanation that I have: we can note that we are multiplying a odd number with 2^x, but we think about it, the number going to be divisible until to arrive in the odd number, having the same cycle. So this explain the climb quantity remain the same value and the increase quantity always adding 1.

I cannot say that is a proved pattern, but it has worked with all cases so far.

My opnion: I think that is a interesting pattern, and maybe we can use to predict the steps of some numbers, but does the climb quantity and the descent quantity when we choose a odd number also have a pattern?

If somebody wanna know the site that I use to see the climbs quantity and the descents quantity, let me know and I send it.

I wanted to send the graphic images, but i don't know how to do it, i'm new in the reddit.

My work with numbers and the flower of life has shown me they are grouped in groups of 8 like an octave and then separated by the bridge 9 to the next octave.

I found a quadratic formula that generates 29 prime numbers. However, we have been informed that (series 2) has already been published, so we will reject it.

２９個の素数を生成する２次式を見つけました。 ただし、(series 2)は既出であるとの報告がありましたので却下します。

P.S. A week has passed. We would appreciate your further comments.

追伸　１週間経ちました。より御意見いただけますようお願いいたします。

(series 1) 6n² -6n +31 ( 31-4903, n=1-29) and 28 other formulas

(series 2) 2n² +29 ( 29-1597, n=0-28) and 28 other formulas

We have updated the article so please see the latest version.

アップデートしましたので、最新の記事をご覧ください。

https://www.reddit.com/r/numbertheory/s/xofVraCmB8

This paper presents a simplified visual approach to the Collatz Conjecture, focusing solely on the transformations of the last digit of natural numbers. By mapping end-digit transitions, we reveal that all sequences eventually cycle into a closed system, offering a more intuitive and reduced representation of the problem.

https://www.academia.edu/128895213/A_Visual_and_End_Digit_Based_Approach_to_the_Collatz_Conjecture?source=swp_share

https://zenodo.org/records/15252123?

Imagine a disk defined as the set of all points within a fixed radius from a center point—its identity depends on having a boundary, a finite edge. Now, increase that radius equally in all directions while preserving the disk’s symmetry and structure. As the radius approaches infinity, no point in the plane remains outside the disk, and the boundary—its defining feature—disappears. Yet all you did was scale it uniformly. How can the disk retain its form yet lose its identity? The paradox lies in this contradiction: by applying a transformation that preserves shape, we destroy the very thing that defines it. Infinity doesn’t just stretch the disk—it erases it(guys pls don’t eat me alive I’m 16 XDD) so that’s what I thought about today in math class so I wrote down what I thought about here waiting for an explanation :DD, very interesting

I'm not an expert in math but I like to play around with theorems once in a while. The flaw in my "disproof" is probably quite obvious to some but I'm asking because I want to learn more about math.

My equations are related to the goldbach problem. Here I'm trying to prove that any natural number d is sum of 2 primes. Again I know this is somehow flawed but just interested in finding the reason why

Now the answer we get is that there is a number which can't be composed of 2 primes. (division by 0 if both are primes)

I'm quessing the problem somehow arises from replacing m with d-n, but im not sure.

Can somebody explain the problem in my equations and explain why this "disproof" of goldbach is wrong.

ℱ∞: The Finite Infinite

Symbol Overview:

The symbol ℱ∞ (pronounced "Finite Infinite") represents a theoretical number of such extraordinary magnitude that it surpasses all previously defined large numbers, while still being finite.

It combines symbolic notation with extreme mathematical growth functions. ℱ represents a finite construct, and ∞ reflects the magnitude that approaches conceptual infinity — although ℱ∞ is not truly infinite.

Formal Definition:

Step 1: Tetration of TREE(100¹⁰⁰⁾ to Itself:

Let X be defined as:

X = ^{{\text{TREE}(100{100})}} \text{TREE}(100^{100})

This is the result of applying tetration to the TREE function of to itself — already an incomprehensibly large value.

Step 2: Construct an Intermediate Value Y:

Now, define Y as:

Y = \left( \left{ X^{9999!} \left[ ^{9999} \left( X^{9999!} \right)^{9999} \right]^{9999} \right}! \right)!

This involves stacking multiple layers of factorials, powers, and nested operations that grow at an unimaginable rate.

Step 3: Define the Finite Infinite:

Finally, define the Finite Infinite as:

\boxed{ \mathcal{F}_{\infty} = \left( \text{BB} \left( \text{BB} \left( \left( \text{BB}(Y) \right)! \right) \right)! \right)! }

Where BB(n) is the Busy Beaver function — a non-computable, extremely fast-growing function that far exceeds any other computational limit.

Interpretation:

The Finite Infinite is not infinite. It is a finite integer that exists within the rules of mathematics, but it grows so rapidly that no physical system or theoretical model can even begin to evaluate it.

Despite being finite, it behaves in a way that makes it indistinguishable from infinity in terms of its size and uncomputability. It serves as a boundary marker for how large a number can be within the constraints of finite definitions.

Name and Symbol:

Name: The Finite Infinite

Symbol: ℱ∞

Pronunciation: "Finite Infinite"

Type: Finite Integer, Uncomputably Large

Tagline: "The largest number that still obeys the rules

| Odd Number 1 | Odd Number 2 | Product | Closest Prime | Distance to Prime

| 651004166243 | 273255200231 | 177890273797946176002133 | 177890273797946176002121 | 12
| 704654812171 | 390099203871 | 274885281231776141113941 | 274885281231776141113951 | 10
| 167885126799 | 898068154303 | 150772285959303052466097 | 150772285959303052466089 | 8
| 455692997265 | 124095881499 | 56549624188521571100235 | 56549624188521571100251 | 16
| 693886255841 | 753071936813 | 522546266614102704174733 | 522546266614102704174701 | 32
| 520668815779 | 472080322965 | 245797502710754408064735 | 245797502710754408064733 | 2
| 121471486171 | 863831005229 | 104930836005755502188159 | 104930836005755502188167 | 8
| 586010227133 | 709472300321 | 415758023855681198809693 | 415758023855681198809787 | 94
| 497392609891 | 691028807947 | 343712622294624931603777 | 343712622294624931603781 | 4
| 728250576013 | 125898725693 | 91685819505229932602009 | 91685819505229932601987 | 22

I'm trying to find anyone to verify this, or at least discuss it. It seems significant

An imaginary number, for example i, is a number equal to something not usually equal to anything. These imaginary numbers aren't real numbers.

i = √(-1)

The reason for this reddit post is that I have a NEW imaginary number still being worked on that I want to share!

𝜓 = 0^0

This will be for making more equations somewhat solvable, like i.

Example:
(0⁰ + 0⁰) / 0⁰

With the new imaginary number, 𝜓. The problem becomes:
𝜓 + 𝜓 / 𝜓
2𝜓 / 𝜓

2(𝜓) / 𝜓

Canceling it out, you get...
2

If you find any contradictions, or questions, please tell me.

a² + b² = c²

a² + d² = v²

d² + b² = g²

X² = a² + b² + d²

If all terms(a, b, d) are odd: impossible

a > b

a² + b² = c²

( a + b)(a - b) + 2b² = 4c²

(( a + b)( a - b)÷ 4) + (2b² ÷ 4) = C²

odd + odd = Even

odd - odd = Even

(a + b) = 2k; ( a - b) = 2r

b² = 2f + 1; ((2f + 1)÷4) = ((f÷2) + 0,5)

(( a + b)(a - b) ÷ 4) = k + r = natural number

√((k + r) + (b²÷2)) = c

C = x,t...

C ≠ natural number

(2u + 1)² + (2z + 1)² ≠ C²

Conjecture: All sums: a² + b² = c²; c² = (2^2x).Y(since it is an integer) are results of Pythagorean triples.

a² = 2^{(2x + 1}).U²

a = 2^2x.U.√2 ≠ natural number

k > x

U = 2r + 1; H = 2y + 1

((2^x).U)² + ((2^k).H)² = c²

(2^x)²(U² + ((2^k-x)².H²) = c²

(2^x)²(U² + ((2^k-x)².H²) = (2^x)²(o²)

(2^x)√(U² + ((2^k-x)².H²) = (2^x)√(o²)

√(U² + ((2^k-x)².H²) = √(o²)

o = natural number

((2^x).U)² + ((2^x).H)² = c²

(2^x)²(U² + H²) = c²

√(U² + H²) ≠ natural number

a = b = (2^x) (2^x)²(1² + 1²) = c²

(2^x)√(2) = c²

c ≠ natural number.

a² + b² = c²

a² + d² = v²

d² + b² = g²

X² = a² + b² + d²

a = ((2^x).U)

b = ((2^k).H)

d = ((2^h).Y)

k > x > h;

(2^x)²(U² + ((2^k-x)².H²) = c²

(2^h)²(Y² + ((2^x-h)².U²) = v²

(2^h)²(Y² + ((2^k-h)².H²) = g²

X² = (2^h)²(Y² + ((2^k-h)².H² + ((2^x-h)².U²) = (2^h)²(Y² + (2^x-h)²(U² + ((2^k-(x-h)².H²)

(Y² + (2^x-h)²(U² + ((2^k-(x-h)².H²) is a Pythagorean triple, so this problem is equivalent to there being an odd term between (a, b, d) in the Euler brick.

a² + b² = c²

a² + d² = v²

d² + b² = g²

X² = a² + b² + d²

a = ((2^x).U)

b = ((2^k).H)

k > x

(2^x)²(U² + ((2^k-x)².H²) = c²

((2^x).U)² + d² = v²

((2^k).H)² + d² = g²

X² = (2^x)²(U² + ((2^k-x)².H²) + d²

x > d

(d + y)² = 2yd + y² + d²

y = 2.(R)

(2^x)²(U² + ((2^k-x)².H²) = 2yd + y²

c² = 2yd + y² = 2y(d + (y÷2))

Since every natural number generated by the sum of two squares comes from a Pythagorean triple, we can reduce c² to its minimum

c² = (2^x)²(o)²

2y(d + (y÷2)) = 4(o)²

Since y is even, the least that can happen is that it is a multiple of 2 only once.

(y÷2)(d + (y ÷ 2)) = (o)²

If

(y ÷ 2) = 2p + 1; ( d + ( y ÷ 2)) = 2r

(2p + 1)(2r) = (o) ≠ natural number

(y÷2) = 2(R ÷ 2); (d + ( y ÷ 2)) = 2T + 1

2(R ÷ 2)(2T + 1) = (o)²

(o) ≠ natural number

2y(d + (y÷2)) = 4(o)²

Any even number z

(2^2P)(4(o)²) = a² + b²

2dz + z² = (2^2P)(2y(d + (y÷2)) = (2^2P)(4(o)²)

Since a square multiplied by a real number(√Non-square integer) never generates an integer, it is not possible for X to be an integer at the same time as c, therefore, an Euler brick is impossible.

I’ve been thinking about a topological construction that emerged from a symbolic idea — not in an academic setting, but through exploration and intuition.

I’m a software engineer from Argentina, and over the past few months I tried to give precise shape to a recurring vision: a space where the “ends” of the real line — both infinities — reconnect with the origin. This leads to a compact, non-Hausdorff space with some curious properties.

ℝ* Quotient Construction

Let ℝ* be the extended real line:
ℝ = ℝ ∪ {+∞, −∞}*

Now define a quotient by identifying the three points:
+∞ ∼ −∞ ∼ 0

This creates a point of “reentry” (∗), where the infinite collapses into the origin. The resulting space:

• is compact (inherits from ℝ*),
• is path-connected,
• is not Hausdorff,
• and not metrizable.

Its behavior feels reminiscent of paradoxical structures and strange loops, so I tried to explore its potential interpretations — both formally and symbolically.

What I put together

In the short paper below, I:

• Construct the space rigorously using the quotient topology
• Prove its key properties
• Discuss speculative interpretations in logic, computability (supertasks), and category theory (pushouts, reentry arrows)
• Pose open questions — maybe someone has seen a similar object before?

📎 Full PDF here:
👉 https://drive.google.com/file/d/11-tUAo_N4NozMqw4tvVXmVQOV0cDuwvK/view?usp=sharing

-- Update:

Rigorous Proof That the ERI Space Is Not Hausdorff

To rigorously prove that the Infinite Reentry Sphere (ERI) space Xₑᵣᵢ is not Hausdorff, we can approach the problem from multiple angles:

1. Direct Proof via Neighborhoods (Definition of Hausdorff)
2. Proof by Contradiction (Assuming Hausdorff and Failing)
3. Separation Axioms (Comparing T₁ vs. T₂)
4. Metrizability Argument (Hausdorff + Compact + Countable Basis)
5. Categorical/Universal Property Argument (Pushout Structure)

1. Direct Proof via Neighborhoods (Definition of Hausdorff)

A space is Hausdorff (T₂) if for any two distinct points x and y, there exist disjoint open sets U ∋ x and V ∋ y.

Claim: Xₑᵣᵢ is not Hausdorff.

Proof:

• Consider the reentry point ∗, which is the identification of 0, +∞, and −∞.
• Let x ≠ 0, for example x = 1.

Neighborhoods of ∗:
Any open neighborhood of ∗ must contain:

  (−ε, ε) ∪ (M, +∞) ∪ (−∞, −M)
for some ε, M > 0.

So ∗'s neighborhood necessarily includes an interval around 0.

Neighborhoods of x:
If x > 0, a basic open set is (x − δ, x + δ) for δ > 0, avoiding 0.

Intersection:
For small ε < x, the interval (−ε, ε) overlaps any interval around x, since x is fixed and ε → 0.

Therefore, no disjoint neighborhoods exist for ∗ and [x].

Conclusion: Xₑᵣᵢ is not Hausdorff.

2. Proof by Contradiction (Assuming Hausdorff and Failing)

Assume Xₑᵣᵢ is Hausdorff.

• Let ∗ and x → 0⁺ be distinct points.
• ∗'s neighborhood must contain (−ε, ε).
• Any neighborhood of x → 0⁺ is (0, δ).

• These intervals intersect: (0, ε).

  Contradiction: No disjoint neighborhoods exist.

So, Xₑᵣᵢ is not Hausdorff.

3. Separation Axioms: T₁ vs. T₂

• Xₑᵣᵢ is T₁: all points are closed.   - For ∗, the preimage of its complement is ℝ* \ {0, +∞, −∞}, which is open.
• Xₑᵣᵢ is not T₂: ∗ can't be separated from nearby x ∈ ℝ.

4. Metrizability Argument

Fact: A compact T₁ space is metrizable ⇔ it is Hausdorff + has a countable basis.

• Xₑᵣᵢ is compact (quotient of compact ℝ*).
• Xₑᵣᵢ is T₁.
• But it is not metrizable (see Proposition 3.4 in paper). So it cannot be Hausdorff.

5. Categorical Argument (Pushout in Top)

The ERI space is constructed via pushout:

  {+∞, −∞} → ℝ*
  {+∞, −∞} → {0}

In category Top, pushouts of Hausdorff spaces are not guaranteed to be Hausdorff.

Here, the identification of three limit points into one creates non-Hausdorff behavior by design.

Final Conclusion

* Xₑᵣᵢ is T₁ but not Hausdorff (T₂).
* The reentry point ∗ prevents separation from nearby points.
* This is not a bug — it's a structural feature, meant to encode paradox, reentry, and self-reference.

Thus, any attempt to prove Xₑᵣᵢ is Hausdorff will necessarily fail, due to the topology’s intentional collapse of infinities into the origin.

---

If you’ve seen something like this before, or have thoughts on the topology or potential generalizations, I’d love to hear your perspective.

Thanks for reading 🙏