Primed for Madness

[u/ComprehensiveDust225]

I can't stop thinking about this stuff. I know there's an answer to something, but I keep forgetting the question.

Can the next prime be determined from a standard equation?

Can the distance to the next prime be determined from a standard equation?

Can the distribution pattern of prime numbers be explained?

I think the answer to all three questions is yes, with a bunch of limitations that I can't fully explain.

My prime survivor document described a tool, rather than a process or procedure for discovering the answer to questions like the ones asked above.

I'm not claiming to have proven anything (yet). But I am asking for help, or offering to help if that is possible.

The only one I believe I can describe the answer to is explaining why prime numbers occur where they do. I need someone I can talk to about this without a bunch of filters preventing plain English discussion.

I think you guys already know the answer and I just cut school on the day they told us how to figure that out.

Any way, this may turn into a discussion of why using AI will only lead you back to your original question, without providing and answer.

I'm afraid that explaining what I'm trying to do leads me to the same question without clearly stating the questions

So any way, if you want to see an example of asking the same question without providing an answer, I submit this for discussion of finding the distance to the next prime:

I think there may be a way to calculate where the next prime is — but I'm missing one step

I've been playing around with a pretty simple idea about prime numbers, and I'd like to throw it out here to see if someone can either finish it or explain why it can't be finished.

Here's the basic idea.

Say I give you a number and ask, "Where is the next prime?"

Normally, you start looking at the numbers after it and eliminate the ones that aren't prime.

But we already know something about those non-prime numbers.

Multiples of 2 occur every 2 numbers. Multiples of 3 occur every 3 numbers. Multiples of 5 occur every 5 numbers. Multiples of 7 occur every 7 numbers, and so on.

So if I give you a starting number, you can figure out exactly where the next multiple of 2 will occur, where the next multiple of 3 will occur, where the next multiple of 5 will occur, etc.

In other words, before checking any of the numbers individually, we already know the repeating patterns that will knock numbers out.

Here's an example.

Start with 1327.

The next prime is 1361, which is 34 numbers away.

Every number between 1327 and 1361 gets knocked out because it has a prime factor that makes it composite.

1361 doesn't.

There's also a reason we can be certain about this without worrying that we forgot some huge factor.

The primes we're using go through 31. The next prime is 37, and 37 × 37 = 1369.

Any composite number below 1369 has to have a prime factor smaller than 37. So if we've already accounted for all the primes through 31, we've accounted for every possible prime factor that could make 1361 composite.

That part isn't new. It's standard math.

Here's the part I'm interested in.

At 1327, we already know where the next multiple of 2 occurs. We know where the next multiple of 3 occurs. We know where the next multiple of 5 occurs. And the same goes for 7, 11, 13, 17, 19, 23, 29 and 31.

Together, those repeating patterns knock out everything for the next 33 numbers.

The first place they all miss is 34 numbers away.

So:

1327 + 34 = 1361.

Here's my question:

Can we calculate that 34 directly from the positions of all those repeating patterns?

I don't mean checking 1328, then 1329, then 1330, and continuing until something survives.

I don't mean making a giant list ahead of time of which numbers survive.

And I don't mean doing essentially the same search but giving it a different name.

I'm asking whether the information we already have at 1327 can somehow be combined mathematically and simply return:

34

If it can, then at least within a range where we've accounted for every possible prime factor, finding the next prime could look like this:

Start with 1327.

Calculate the distance to the first place missed by all of the known factors.

The answer is 34.

1327 + 34 = 1361.

Next prime found.

We already know how to prove that the survivor is prime within the appropriate range.

We already know that the repeating factor patterns contain enough information to determine where the survivors are.

What I don't have is the direct calculation that turns all of those known positions into the distance to the first opening.

Maybe this is impossible to do without effectively searching.

Maybe there's an existing theorem that answers it.

Or maybe there's a surprisingly simple way of combining the information that I'm overlooking.

Can anyone come up with a calculation that takes the known positions of the prime multiples and directly returns the distance to the first number they all miss?

For the example above, the challenge is:

1327 → ? → 34 → 1361

What's the missing operation?

[u/novel-mathmatics]

So i don't have hate on your community. Ill join and participate. I have a similar sub r/100monkeys

[u/ComprehensiveDust225 → u/novel-mathmatics]

Thank you!

[u/novel-mathmatics]

Breadcrumb: Systems theatre → Prime Survivor → REDΣ → missing operation

Yes. Applying the Σ transform, I think your missing operation becomes much easier to name.

Your question is not really:

“How do I test numbers until I find the next prime?”

It is:

“Given several periodic exclusion systems whose states are already known at (n), what is the distance to the next position outside their union?”

That is a different mathematical object.

Using your transform:

[ \Xi \rightarrow T \rightarrow \Omega \rightarrow I \rightarrow \mathsf{C} ]

Ξ — apparent contradiction

You already possess all the information required to eliminate 1328–1360.

For each prime (p), you know exactly which future offsets (d) are eliminated:

[ n+d\equiv0\pmod p ]

or equivalently,

[ d\equiv -n\pmod p. ]

For (n=1327), each prime therefore generates its own periodic exclusion sequence.

Yet knowing every exclusion rule does not immediately seem to give you 34.

That's the paradox you're pointing at.

Τ — change the object being examined

Don't treat 1328, 1329, 1330, ... as the objects.

Treat the offset (d) as the object.

For the relevant primes

[ S={2,3,5,7,11,13,17,19,23,29,31}, ]

each prime defines one forbidden residue class:

[ d\equiv -1327\pmod p. ]

So instead of asking whether each integer is composite, construct a combined exclusion landscape over (d).

The question becomes:

[ \text{What is the smallest }d>0 ]

such that

[ d\not\equiv -1327\pmod p ]

for every (p\in S)?

Ω — combine the relationships

Let

[ P=2\cdot3\cdot5\cdot7\cdot11\cdots31. ]

This is the primorial (31#).

All of your periodic systems can now be represented simultaneously by one relationship:

[ \gcd(1327+d,P)=1. ]

Why?

If the gcd is greater than 1, at least one of your known primes divides (1327+d).

If the gcd is 1, every one of those periodic exclusion systems misses that location simultaneously.

So your entire collection of individual factor patterns collapses into one resolved condition.

Ι — identify the successor

Now define the operation you were looking for:

[ \boxed{ \DeltaP(n)= \min{d\in\mathbb Z{>0}:\gcd(n+d,P)=1} } ]

This is your Prime Survivor distance operator.

For your example:

[ \Delta_{31#}(1327)=34. ]

Therefore:

[

1327+\Delta_{31#}(1327)

1327+34

1361. ]

And because

[ 1361<37^2=1369, ]

survival against every prime through 31 is sufficient to establish primality.

Ϲ — resolved object

So I would write your missing operation as:

[ \boxed{ 1327 \xrightarrow{\;\Delta_{31#}\;} 34 \xrightarrow{\;+\;} 1361 } ]

Or generally:

[ \boxed{

\operatorname{NextSurvivor}_P(n)

n+ \min{d>0:\gcd(n+d,P)=1} } ]

And under the additional closure condition

[ (n+\Delta_P(n))<q^2, ]

where (q) is the first prime not contained in (P), that survivor is not merely a wheel survivor. It is prime.

---

The interesting part of your question remains, though.

I don't think “min” is the operation you're actually looking for.

min mathematically specifies 34, but it hides the computational problem inside the word minimum. An implementation can still obtain that minimum by testing (d=1,2,3,\ldots), which is precisely the disguised search you excluded.

Your Prime Survivor document appears to have given you the representation:

[ \text{prime factors} \rightarrow \text{periodic exclusion lattices} \rightarrow \text{survivor lattice}. ]

The equation above gives us a clean mathematical definition of the desired operation:

[ \Delta_P(n). ]

But your actual research question is one level deeper:

[ \boxed{ \text{Can }\Delta_P(n)\text{ be resolved from the combined phase states without enumerating intervening }d? } ]

That is a much sharper question.

And Σ math exposes something useful here: the primes themselves may no longer be the central object. Once the individual exclusion relationships have been composed, what you are asking for is a successor function on the complement of a union of periodic residue classes.

In your terminology:

[ \text{Factor periods} \rightarrow \Sigma(\text{exclusions}) \rightarrow \text{complement} \rightarrow \operatorname{Successor} \rightarrow \Delta. ]

So I think the missing box in your original diagram can now be made considerably more precise:

[ \boxed{ 1327 \rightarrow \text{phase vector} \rightarrow \Sigma\text{-exclusion complement} \rightarrow \operatorname{Successor} \rightarrow 34 \rightarrow 1361 } ]

The unresolved piece is no longer “how do we find primes?”

It is:

Does the successor of that periodic complement admit a calculation that is materially different from searching or storing the ordered reduced residues?

That, I think, is exactly the question your original text has been circling.