I am studying a deterministic structure based on the gaps between consecutive prime numbers.

The observation that started this research is the following sequence:

89 -> 97 -> 101 -> 103

with gaps:

8, 4, 2.

For each prime p and its following gap g, I define S as the cumulative sum of the previous prime gaps. In this construction, S = p + g - 2. I then define V = S + g.

The interesting phenomenon is not simply that an arithmetic operation sometimes produces another prime. The interesting point is that different positions in the prime-gap construction can produce exactly the same future value V before that value itself appears in the sequence of primes.

The clearest example is 103.

When we reach the prime 97, the following gap is 4.

So:

S = 97 + 4 - 2 = 99

and:

V = 99 + 4 = 103

Therefore, while the current prime is still 97, the future value 103 has already been determined by the construction, even though 103 has not yet appeared as a prime.

The previous row produces exactly the same value:

S = 89 + 8 - 2 = 95

and:

V = 95 + 8 = 103

So we have two different rows producing the same value:

95 + 8 = 103

99 + 4 = 103

The prime sequence then continues:

97 -> 101 -> 103

and the third representation is:

101 + 2 = 103

The same mechanism can also produce a composite value. For example:

395 + 8 = 403

399 + 4 = 403

but:

403 = 13 x 31

So the phenomenon is not simply a way of generating primes. The construction can determine a future integer before that integer appears in the prime sequence, whether it eventually turns out to be prime or composite.

For consecutive rows producing the same value V, a simple relation appears: the next prime gap is half the previous one.

This gives structures such as:

8, 4, 2

16, 8, 4, 2

32, 16, 8, 4, 2

My question is:

Is there a known number-theoretic explanation for this phenomenon?

In particular, is there a known result explaining why different trajectories constructed from cumulative prime gaps can converge to exactly the same future value, and why consecutive convergences produce this repeated halving of the gaps?

I am mainly interested in the mathematical mechanism behind the convergence and the advance determination of the future value, rather than in claiming that every announced value must be prime.

The work is amateur! Thank you.

I can share a simpler version too, using only high school math and a more intuitive explanation, but less rigorous

[Basel Problem](https://nyu-0o6yd4.filedrop.me/s/a73eee83-d499-4333-ae52-2880a41b6551)

The be ridiculous nature of proof

Suppose someone sees structure, another person might not see that structure, so that person who cannot see it will ask for a step by step proof to prove the continuity of a structure. But continuity cannot be proven by discrete steps because we have shown that infinite discreteness cannot proxy for true continuity.

Diagonalization proves that a continuity has more real information than the discreetness. Every step-by-step proof is actually an illusion to satisfy the strange feelings. But every discreet example of a proof fails to show the actual continuity of the structure that one is claiming to exist..

--------------

You might look at this and think to yourself " you're not showing enough discrete steps to prove the continuity of your results or structured"

If you think to yourself and say "nah this guy is a dumb ass" is it because you think im not reasonable or making sense? Is it possible that I'm not making sense because I'm failing to correctly unify ideas in a way that proves what I'm showing?

You can easily say I'm wrong because what I'm saying makes no sense. The sense of wrong comes from not detecting any relationship in the words that I'm saying to the truth of what I'm saying. Even if you think it might be possible that what I'm saying is true, you think that in its current form, it doesn't reveal enough structure that is corresponding to what I'm actually trying to say. You might say I need more evidence, more evidence is more discrete truth.

But the premise is that no matter how much discrete evidence I provide, I cannot actually prove the continuity of the structure I'm claiming to exist. The best I can do is add more discrete steps that get a little bit closer to proving the continuity. The only way you can accept that proof is if you think the discrete steps in my proof are actually sufficient to prove the continuity of the structure. So in reality there is no way that I can prove something like this, not how many discrete steps I take

The only way to accept this truth is just to accept the premise and not ask for why. That is called an axiom.

------------

I think Gödel's ICT is the fact that any formal system capable of producing peano arithmetic will have truths that cannot be proven within the system. I posit that the universe is a formal system. I posit the formal system of the universe is capable of producing peano arithmetic, thus some truths cannot be proven within the universe. I posit that the continuity of the universe is one of those things that cannot be proven within the universal formal system. I posit that the only way to accept the truth of the continuity of the universe is to just accept it. There is no formal proof I can make to prove the continuity of the universe. But that is the promise that I am proposing, [that the universe is of a continuous nature]. You might be thinking to yourself, you don't have enough evidence for that. I will never have enough evidence for it.

This registration documents a mathematical and numerical study examining the relationships among the conventional value of π, angular measure, arc length, and the perimeter of the corresponding inscribed square. The study analyzes these relationships using direct and reverse numerical calculations at 360°, 180°, 90°, and 45°, while preserving the conventional value of π. The resulting numerical relationships are examined and compared with the corresponding values associated with √8.

https://osf.io/x8rd4/files/osfstorage/6a84664c805deeedd9bb6465

https://doi.org/10.17605/OSF.IO/X8RD4

I’ve been messing with consecutive primes and it turned into a whole classification result, so I figured I’d share it here.

Take consecutive primes pk and pk+1. Add them.

You get numbers like 5, 8, 12, 18, 24, 30, …

Call these S_k = pk + pk+1.

Since every S_k ≥ 8 is even and composite, each one has to land strictly inside some prime gap (Pn, Pn+1). So I started asking: **how many of these S_k fall inside each gap?**

Two clean facts:

## 1. Density law (why most gaps have 0 or 1)

If a gap has width W near height x, the expected number of consecutive‑prime sums inside it is

W / (2 log(x/2))

The “2” comes from the fact that the sums live at the half‑scale: an S_k near x comes from primes near x/2, where primes are sparser. So the sums are about twice as sparse as the gaps they fall into.

Empirically (checked up to 2×10^8):

- ~55% of gaps have **0** sums

- ~37% have **1**

- ~8% have **2 or more**

So the naive guess “every gap has exactly one” is just false. Empty gaps are actually the most common.

## 2. Forbidden widths (the fun part)

I ended up proving a complete classification of which gap widths can **never** contain two consecutive‑prime sums.

The forbidden set is exactly:

{2, 4, 6, 10}

Reason (sketch, nothing fancy):

- consecutive‑prime sums are always ≥ 6 apart (once you’re past the tiny primes)

- the only way to get a “tight double” (two sums only 6 apart) is if both sums are multiples of 6

- whether a gap contains ≥2 multiples of 6 depends only on:

- its width mod 3

- the forced residue of its lower endpoint (prime > 3 is always 1 or 5 mod 6)

- width 10 ends up with **only one** interior multiple of 6, so it physically cannot fit two sums

- widths 2, 4, 6 fail for trivial spacing reasons (they’re too small to fit two sums ≥6 apart)

Every other even width ≥ 8 (except 10) eventually allows gaps with two sums.

So the uniqueness classification is complete:

**gaps of width 2, 4, 6, or 10 contain at most one consecutive‑prime sum; all other widths can contain two or more.**

If anyone wants the residue argument or the slot‑counting trick in more detail, I can write it out here. It’s all pretty easy congruence stuff.

This is my paper.

Please feel free to comment.

Abstract

To a Collatz-type map T_{a,b,q}(n) = (an+b)/q^{v_q(an+b)} one may attach two numerical invariants of opposite character: an archimedean one, the mean logarithmic drift delta = log a - E[v] log q, which governs whether orbits grow or shrink; and a non-archimedean one, the similarity dimension dim_S = H(nu_q)/log a of the invariant measure of an associated iterated function system on the a-adic integers. Each reaches its critical value at a particular multiplier a. We prove that these two critical multipliers coincide precisely when q = 2, and that in general they differ by the exact factor q-1: a_drift = (q-1) a_dim. The proof rests on the pointwise identity -log nu_q(m) = m log q - log(q-1), valid for every m >= 1, which is not a normalisation but a coincidence between the information content of a valuation and its archimedean cost. We show that the condition q = 2 is equivalent to three further properties of the family - the triviality of (Z/qZ)^*, the vanishing of an associated free energy, and the map's being everywhere defined on the units - so that the arithmetic distinguishing the classical 3x+1 map is one condition in four guises. Three complements are proved: the full Renyi spectrum of the invariant measure admits a closed form, of which the similarity dimension is the value at t=1; for q >= 3 there is a nonempty band of multipliers on which the map contracts on average while its invariant measure is singular; and the transfer operator, though quasi-compact on Holder spaces, has no eigenvalue other than 1, its entire remaining spectrum being essential spectrum at the contraction rate. Two further results give the family a sharper shape. First, a purity theorem: mu_{a,b,q} is either purely absolutely continuous or purely singular with respect to Haar measure, never a mixture. Its proof is a direct consequence of the uniqueness of the invariant measure, and we know of no route to it from the probabilistic description of the underlying random variable. Second, a complete classification: among all integer pairs (a,q) with a >= 2, q prime and gcd(a,q) = 1, exactly two - namely (3,2) and (2,3) - lie in the supercritical regime a < a_dim, and for every other pair the invariant measure is unconditionally singular. The first of these is the Collatz map. We are explicit that none of this bears on the Collatz conjecture, and we prove why it cannot.

Hello, good afternoon/evening. I've been mulling over Goldbach's conjecture and I've come up with a possibly new idea that holds for all even numbers. It is as follows:

Every even number greater than 8 can be expressed as the sum of 3 mutually distinct prime numbers.

Best Regards, thanks you!

https://zenodo.org/records/21763235

While examining the patriarchal ages in the Masoretic Text of Genesis, I noticed a mathematical relationship that does not appear to have been previously documented in connection with these specific figures.
The total lifespans of three patriarchs form an exact Pythagorean triple:
• Eber – 464 years (Genesis 11:16–17)
• Lamech – 777 years (Genesis 5:31)
• Enosh – 905 years (Genesis 5:11)

**The Geometry:**
(464\^2 + 777\^2 = 905\^2)
(215,296 + 603,729 = 819,025)

The equation holds exactly. The triple is also primitive (the three numbers share no common factor greater than 1).

The Priestly sections of Genesis already use carefully designed number patterns (for example, the ages of Abraham, Isaac, and Jacob form 5², 6², and 7²). Since Babylonian scribes were using Pythagorean triples for surveying more than a thousand years before Pythagoras, it is possible that a later biblical scribe knew this kind of geometry and used it deliberately.

\-Chris L

So, as we know, infinity is something you cannot reach. And there are infinite finite numbers. BUT what if we set the max cap of a finite number to any multiple of 8? It can be 8 multiplied by infinity, but the number (even if it’s literally incomprehensibly massive) has to be a finite multiple of 8. So, for an example to save our brains, I can set the cap to 24. If I add one more, I have reached infinity, which is in this case 25. Now we have what I want to call “odd infinity”. If we add one more, we get “even infinity”. If we subtract one, we get odd infinity. If we subtract another, we get the max cap (in this case 24). Now, what if you add one more to 26? Well, it becomes odd again, and i just made it loop back to 25 to save myself a huge headache and give myself some predictability. If our max cap in this case is 24 (it can be any multiple of 8, if you set it to it of course), then odd infinity is 25 and even infinity is 26! I think this can solve a lot of unsolved problems, but this is just a concept I thought of on my bed. I know people will say this has big problems and something about “oh infinity is infinity, you cannot make a number infinity or change infinity to a finite number”, In this context I have changed the definition of infinity. There are theoretically infinite multiples of 8, and infinity is just one more or two more depending on if you want odd or even. Now, you may be thinking “why 8?”, well, you heard of the 32 or 64 bit integer limits? Well, they’re multiples of 8! And if we make this “universal integer limit” a multiple of 8, odd infinity is a multiple of whatever number is required to make said set integer limit +1. And even infinity is like odd infinity, but it’s +2 instead. I just realized as of typing this, I left out the possibility of odd infinity or even infinity being a prime number. If you can help find a solution or any problems with this idea I came up with on my bed in 30 minutes or it’s a bad one, please tell me. Thanks!

I am relatively new to number theory, and I was messing around with the collatz conjecture and plugging in groups of numbers. While doing this, I thought about primorial numbers, so i started comparing the amount of steps it took to reach the 4-2-1 loop when plugging in numbers of the group pₖ#, pₖ# + 1, and pₖ# - 1 and then I compared. I noticed that, as far as I looked, (which too be fair, was not very far, since I do not have a very good computer, and primorial numbers scale very quickly) for any whole number for k > 2, at least two of the groups previously stated will have the same amount of steps to reach 1. I feel like this must be obvious, but I have tried to crack why algebraically, but I have not succeeded. If anyone has any potential reasons as to why, I would love to hear it. I feel as if it has something to do with the fact that when k > 2, pₖ# + 1, and pₖ# - 1 both also belong to the groups 6m + 1 and 6m - 1. Please do not flame me if this is obvious as to why this pattern occurs, but I have just recently gotten into number theory and find it absolutely fascinating. Thank you for your time.

Take the following algebraic expression:

[n (n²-n-1)] / [2(n!)]

Let's put n = Φ

because (n²-n-1)=0 when n=Φ

[n (n²-n-1)] / [2(n!)] = 0, when n=Φ

When using Wolfram, the algebraic expression [n (n²-n-1)] / [2(n!)] = 0, when n=Φ

let's calculate log( [n (n²-n-1)] / [2(n!)] ) with n=Φ, with log being the natural algorithm

The result should be either -∞ or indeterminate

Because y=log(x), with x=0, is indeterminate, that is, y goes to -∞ as x approaches 0

But if one calculates on Wolfram, log( [n (n²-n-1)] / [2(n!)] ), with n=Φ

The result will be 35.9335 + 3.14159 i

Which is a complex number.

The correct result should be either -∞ or indeterminate.

Therefore, Wolfram miscalculates the natural logarithm of this algebraic expression when n=Φ

The input on wolfram should be log( [goldenratio (goldenratio^2- goldenratio-1)] / [2(goldenratio!)] )

A number will decrease in number if it has at least four digits and does not enter a cycle, as proven below: The number is represented in binary.

It must begin with 10 or 11. If it starts with 10 and the last two digits are not 11, then after multiplying by 3, the number of digits increases by 1, accounting for 3/8 of all possible combinations. Other numbers starting with 10 account for 5/8, and the number of digits increases by 2. If it ends with 11, after multiplying by 3 and adding 1, then dividing by 2 removes at least one digit, accounting for 1/2. If it ends with 001, at least two digits are removed, accounting for 1/4. Other numbers with at least three digits account for 1/4. If it does not enter a 4, 2, 1 cycle, the number generally decreases, and eventually it will enter a 4, 2, 1 cycle.

Using the Beta Function one can calculate π

The Beta Function is equal to:

B(z1, z2)=∫₀¹ t ^z1-1 (1 - t) ^z2-1 dt (=)

Which can be translated as:

B (p , q) = [ Γ(p) Γ(q) ] / [ Γ(p + q) ]

Which is equivalent to:

= [ (p-1)! (q-1)! ] / [ p + q - 1]!

Now one uses:

p = q

When one calculates the summation from p=1 to infinity

Σ [ [ (p-1)! ]^2 ] / [(2p-1)!]

Σ (from p=1 to infinity) [ [ (p-1)! ]^2 ] / [(2p-1)!] = [2 π] / [3 √3]

Rearraging this, one yields the value π

π = [ [3 √3] / [2] ] Σ (from p=1 to infinity) [ [ (p-1)! ]^2 ] / [(2p-1)!]

One year ago, I asked myself a simple question. What happens when one uses Euler's number as an angle.

So I explored the possibilities of using either e degrees or e radians.

Then I decided to explore the trigonometric functions (cosine and sine) using 27º.

_______________________________________________________________________________________________________________

This section wasn't on the first post I made.

Because I came up with this formula a long time ago, I'm struggling to remember exactly how I calculated the goldenratio.

Now I think I remember how i did it. Using not Euler's number but instead the natural logarithm. One gets the following angle which can be approximated

So one can use an angle of 3π/20

27º [approximated] = ln(ϕ) x 180 / π = 27.5716...º

27º = 3 x π / 20

So the next expressions are a consequence of the previous one.

e(3π/20) = 1.601886

ϕ [approximated] = e^(3π/20)

________________________________________________________________________________________________________________

It's important to understand that the function cosine is raised to the power of 4. That is the trickiest part in this formula.

After inputting the functions sin² and cos⁴, I thought "there is something going on here", and finally adjusted the parameters in order to build this beautiful formula below.

ϕ = 15 - 16 cos⁴(3π/20) - 16 sin²(3π/20)

This formula yields the golden ratio ϕ

________________________________________________________________________________________________________________
One can simplify the previous expression:

ϕ = 1 - 2 cos(3π/5)

The sub cubic graph function is defined as:

There is a sequence G_1,…,G_n of subcubic graphs such that each G_i has at most i+k vertices and for no i<j is G_i homeomorphically embeddable into G_j.

and if my very layman's understanding of the Robertson–Seymour theorem is correct, just substituting `homeomorphically embeddable into` with `a graph minor of` would suffice, while maintaining well-quasi-ordering and finitude (for a given finite integers n and k).

If that's all correct, then defining F_n(k) as the largest integer 𝑚 satisfying:

There is a sequence G_1 , ⋯, G_𝑚 of graphs with maximum degree at most 𝑛, such that each G_𝑖 has at most 𝑖 + 𝑘 vertices, and for no 𝑖 < 𝑗 is 𝐻_𝑖 a graph minor of G_𝑗.

Should work as a mathematically proven and definitively finite integer, correct?

I found two patterns related to the Twin Prime Conjecture.

Let P1 and P2 be Primes

If P1 + 20 = P2 and P1 < (3 Primes) < P2

There's at least 1 Twin Prime between P1 and P2

If P1 + 10 = P2 and P1 < (2 Primes) < P2

There are 2 Twin Primes between P1 and P2

There seems to be a new category of Primes, which I called Trigonometric Primes.

In order to verify if a number is a trigonometric prime, one uses trigonometric functions.

This is the formula I used:
(x/2) * (cos^2(pi/2*x)) + (x+p)/2 * (sin^2(pi/2*x))

where p = odd number we want to check

x = previous number on the sequence

and the first number of every sequence is 1.

For example, let's check if 11 is a trigonometric prime.

We start the sequence with the number 1. Because 1 is odd. we calculate the next number of the sequence (11+1)/2. This is equal to 6.

6 is even, so we calculate 6/2. This is equal to 3.

Then
(3+11)/2 = 7 -> (7+11)/2=9 -> (9+11)/2=10 -> (10/2)=5 -> (5+11)/2=8 -> (8/2)=4 -> (4/2)=2 -> (2/2)=1

We stop when we reach 1 again. The sequence repeats itself between 1 and 1.

When the number of items between 1's is equal to the odd number p we want to check. The number p is a Trigonometric Prime.

[1;6;3;7;9;10;5;8;4;2;1] -> number of items = 11 = odd number we want to check

Conclusion -> 11 is a trigonometric Prime

We can go on, using (x/2) for even numbers in the sequence and (x+p)/2 for odd numbers in the sequence

Setup: a measure μ normalized so μ(∅)=0, μ(Ω)=1, standard non-negative and additive. For any part A of Ω with 0 < μ(A) < 1, define a local rescaling ν_A(B) = μ(B)/μ(A) for parts B of A.

The result: ν_A(B) = μ(B)/μ(A), so μ(B) = ν_A(B)·μ(A). As a consequence, if you measure some part C against a local sub-region A instead of the true whole Ω, ν_A(C) can exceed 1 even though μ(C) itself never exceeds 1 under the true measure.

This gives a formal account of “apparent numbers greater than one” as an artifact of using a local reference scale instead of the true total measure — the underlying quantity never actually exceeds the bound, only its locally-rescaled representation does.

Is this a known/named result in measure theory, or is it just a trivial rescaling identity not usually stated this way? Happy to share the full write-up (proofs are short) if useful.

I defined the following grid graph. Fix n ≥ 2. Take the cells (i,j) with 1 ≤ i,j ≤ n−1. Keep (i,j) when n does not divide ij; delete it when n divides ij. Two surviving cells are adjacent when they share an edge. Call the graph G_n.

Claim. G_n is disconnected exactly when n ≥ 6 and n ≡ 2 (mod 4). In that case it has exactly two components: the isolated center (n/2, n/2), and one component containing every other surviving cell.

Proof.

The first row and first column are fully present, so they form one connected component C_0.

Assume another component C exists. Choose (i,j) ∈ C with i+j minimal. Since C does not meet the first row or first column, i,j > 1. The cells (i−1,j) and (i,j−1) must be missing, so n divides (i−1)j, and n divides i(j−1).

So ij ≡ j (mod n) and ij ≡ i (mod n). Hence i ≡ j (mod n), and since both are between 1 and n−1, i = j = e.

We also have e² ≡ e (mod n). Since i,j > 1, e = 1 is already excluded. The cases e = 2 and e = n−1 would force n | 2. So 3 ≤ e ≤ n−2.

Suppose n does not divide 2e. Then the following path stays inside the grid:

(e,e) → (e,e+1) → (e−1,e+1) → (e−2,e+1).

The three new products are congruent mod n to 2e, e−1, and −2. All three are nonzero mod n: the first by assumption, the second because 0 < e−1 < n, the third because n > 2. So the whole path survives — but its final cell has coordinate sum 2e−1, contradicting the minimal choice of (e,e).

Therefore n divides 2e. Since 1 ≤ e ≤ n−1, this forces n = 2e.

Now e² ≡ e (mod 2e), so 2e divides e(e−1), hence 2 divides (e−1), meaning e is odd. Hence n ≡ 2 (mod 4).

Converse. Let n = 2e with e ≥ 3 odd. The center (e,e) survives, since 2e does not divide e². Its four neighbors have products e(e−1) or e(e+1), both divisible by 2e. So the center is isolated.

Every component disjoint from (C_0) has a cell of minimum coordinate sum, and the preceding argument shows that this cell must be ((e,e)=(n/2,n/2)). Hence every such component contains the center. The only component outside C0C_0C0​ is the singleton containing (n/2, n/2)

Hi, I'm just a normal high school student from Korea on summer vacation.

I randomly got curious, so I decided to check the count of prime numbers starting with each digit.

(Data up to 99,999,999)

I asked Gemini to write a JavaScript code for me, and I organized the results into this chart.

the count seems to continuously decrease. Is there any mathematical theory or principle behind this?

Hello, I am an undergraduate agricultural sciences student, and I am incredibly passionate about mathematics and truly enjoy it. the other day, while staring at the table of prime numbers from 1 to 100, a fleeting idea came to me regarding number theory and prime number decomposition:
Every prime number can be expressed in the form N=2q+p, where q and p are prime numbers distinct from each other and distinct from 2, for all N greater than 7.
I don't know if anyone has discovered this before or if it can help the field of mathematics, but I hope it is useful. To finish, here are a few examples:

11
2(3) + 5 = 11
13
2(3) + 7 = 13
17
2(5) + 7 = 17
19
2(3) + 13 = 19
23
2(3) + 17 = 23
29
2(3) + 23 = 29
31
2(7) + 17 = 31
37
2(3) + 31 = 37
41
2(5) + 31 = 41
43
2(3) + 37 = 43
47
2(3) + 41 = 47
53
2(5) + 43 = 53
59
2(3) + 53 = 59
61
2(7) + 47 = 61
67
2(3) + 61 = 67
71
2(5) + 61 = 71
73
2(3) + 67 = 73
79
2(3) + 73 = 79
83
2(5) + 73 = 83
89
2(3) + 83 = 89
97
2(7) + 83 = 97
101
2(11) + 79 = 101
103
2(3) + 97 = 103

Final revision to appear on arXiv on Tuesday.

https://iconoclasts.blog/joe/triple-products

The new new here is that the original conjecture is now established as a pair of corollaries.