r/numbertheory

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..

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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.

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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.

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.

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.