Prime number theories

[steady robin]

So here I'll put some of my little thoughts and stuff about prime generating/ prime checking stuff but mainly prime generating polynomials

Made ts cuz math discussion was getting flooded by my useless yapping

Anyway

So first one is
( L(n) - n² ) + N

Lemme try it out a bit

Aight it's broken

Now I'm js coming up with smth

24 * (L(n) - n²)

[torpid ridge]

It's impossible to have a prime-generating polynomial.

It's provably impossible.

[native sandal]

what is L(n)

[steady robin]

Nth Lucas number

[steady robin]

Maybea

U never know

[native sandal]

no

its proved to not exist

[steady robin]

It's only proved that it's probably robably impossible

[native sandal]

there is no probably impossible...

[steady robin]

U can't prove it's impossible

[steady robin]

• we are already close to making one

We only need to reverse a equation

[native sandal]

sure mate.

👍

[steady robin]

It doesn't have to generate primes consistently

The congruence

[native sandal]

ahhhh so something like 6n - 1?

[steady robin]

Yeah but it has Lucas numbers

Wait u gave me a idea but it's probably not going to work

[steady robin]

So it's about the congruence L(n) = 1 ( mod n)
Pretend that the equals has 3 lines

It isn't unique to the primes, but it will always hold true for the primes

[native sandal]

Ohhh so you are not deriving it?

just testing random formulas till it works

[torpid ridge]

The Lucas numbers aren't generated by a polynomial.

[torpid ridge]

Yes it does, otherwise what's the point?

If that's your standard, the "polynomial" y = x literally generates every prime.

[native sandal]

I mean here is better formula that what he will give 2n -1 has all primes greater than 2

😭

[steady robin]

Yes lol why i made this, cuz I didn't want to fil math discussion with yapping lol

[steady robin]

More consistently than that

[steady robin]

No bro,I said L(n) mod n = 1

[steady robin]

Yal do I delete this discussion nun is happened anyway

[torpid ridge]

Except you just said consistency isn't a factor.

[torpid ridge]

I know that's what you said. I said that any prime-generating function including the nth Lucas number isn't a polynomial, because the function that generates the Lucas numbers isn't a polynomial.

[thin berry]

If the polynomial is degree Infinity it's possible

[torpid ridge]

A "polynomial" of "degree infinity" isn't a polynomial.

[thin berry]

well yeah, but I think they meant that in what they thought polynomial meant

[torpid ridge]

No, they think the Lucas number function is a polynomial.

[thin berry]

no I'm saying I think their definition of polynomial is wrong

[torpid ridge]

I agree with that much.

[thin berry]

By a definition that includes "degree Infinity", the Lucas numbers are a polynomial

or rather, they have a polynomial that is equal to them at integer values

ofc there's infinitely many polynomials that satisfy that

[torpid ridge]

I imagine there's some equivalent of Binet's formula for the Lucas numbers, which is exponential, but expressible in terms of the Taylor series of e^x.

[thin berry]

I believe it's $(\frac{1+\sqrt{5}}{2})^x+(\frac{1-\sqrt{5}}{2})^x$

[knotty ferry]

Coffey 2.0

[jade wave]

we literally CAN

we DID

[thin berry]

There's a way to make an infinite polynomial that goes through any countable set of numbers that pass the vertical line test, I think

[thin berry]

$g_n(x)=g_{n-1}(x)+(Y_n-g_{n-1}(X_n))\prod_{k=0}^{n-1}\frac{x-X_k}{X_{n}-X_k}$

$g_0(x)=Y_0$

Here it is

[knotty ferry]

Coffey 2.0

[thin berry]

$\lim_{n \to \infty} g_n(x)$ is a polynomial for some set of points $(X_0,Y_0),(X_1,Y_1)...$

[knotty ferry]

Coffey 2.0

[jade wave]

and whats the degree of g_n unless (Xi, Yi) are already a set of form (Xi, P(Xi)) for some polynomial P?

[thin berry]

g_n is at most degree n

it's some integer between 0 and n

If there's only k points and not infinite then
g_(k-1)(x) works

https://www.desmos.com/calculator/0dkatn252e

[thin berry]

Oh nice, you can easily see how if two X_n s are equal
(meaning the polynomial would have to have two values at one x value, but that's not a function)
then it doesn't work, bc then you end up dividing by 0

[thin berry]

Oh I think they mean polynomials that sometimes generate primes

They were noticing how 6k+1 returned a lot of primes the other day, I'm assuming they mean like a+bk

[native sandal]

then what he is doing is useless

there is already a well known theorem

for it

[torpid ridge]

y = x returns literally every prime.

[thin berry]

yeah

[thin berry]

they're doing it for fun I think

[torpid ridge]

"Polynomials that sometimes return primes" are trivial and of no serious interest. "Polynomials that always return primes" would be of great interest if they existed, but they provably do not.

[thin berry]

yeah it's not serious interest

[thin berry]

^

[torpid ridge]

I don't understand why it's any kind of interest in something so trivial.

[thin berry]

It's not trivial to them

they're learning

and playing around with numbers

why stop them?

they're just having fun

[torpid ridge]

Are they? They haven't even stated any concrete condition or conjecture they could be investigating.

[thin berry]

^

[torpid ridge]

Okay, but what is a "prime generating polynomial"?

[thin berry]

^

I shouldn't speak for them though

that's just my guess

[torpid ridge]

They were talking about invoking the nth Lucas number, which isn't a polynomial at all. I don't think they even understand what they're doing, which is why I'm critical.

[thin berry]

Tbh I'm confused bc now I realize they never said the Lucas numbers were a polynomial

I think by "the first one" they meant one of their little thoughts about prime generating/ prime checking stuff, and not a prime generating polynomial

[thin berry]

Oh nvm

ok but, still I think that's not what they were originally saying

[native sandal]

i think they are probably doing test to some large number and say it generates primes

[torpid ridge]

We don't know because they're that bad at saying what they mean, which means they're bad at math.

[steady robin]

Excuse you

I just for norifs from this chat

[steady robin]

Bruh I was just posting little uselles polynomials in this channels it ain't that deep

[torpid ridge]

Except 1) they aren't polynomials if you're using Lucas numbers, 2) you're claiming these are prime generating polynomials without elaborating on what that means.

[steady robin]

So do I say what it means?

[steady robin]

But aren't Lucas numbers generated by a polynomial?

[torpid ridge]

No!

[steady robin]

Like Lucas n is square root of 5 something's

[torpid ridge]

They're generated by an exponential!

[torpid ridge]

No.

[steady robin]

Huh

Wdym no

[torpid ridge]

That's wrong.

[torpid ridge]

No. That is just wrong.

[steady robin]

Why's it wrong

[torpid ridge]

How is it right?

[steady robin]

Look at the screenshot

[torpid ridge]

I did. That doesn't answer my question.

[steady robin]

Just try to use the polynomial

And it will give the nth lyrics number

[native sandal]

🤔

[torpid ridge]

Which polynomial??? An infinite family of them is described!

[steady robin]

The one in the screenshot

The first one

[native sandal]

thats it he said do it

💀

[steady robin]

Lol

[torpid ridge]

That's not a polynomial.

[native sandal]

Verification is trivial given an infinite power source, infinite computational time, and an ideal computing environment.

[steady robin]

Just change n

[torpid ridge]

If n changes it's not a fucking polynomial.

[steady robin]

Euler's prime generating polynomial uses a n

[native sandal]

🤡🤦‍♂️

[torpid ridge]

What the fuck do you think a polynomial is?

[steady robin]

Formula with variable

[torpid ridge]

No.

That's not even a little bit correct.

[steady robin]

Then idk

[torpid ridge]

A polynomial is a function of the form $f(x) = \sum_{n = 0}^k a_n x^n$.

[knotty ferry]

Techie Literate

[native sandal]

maybe losen up the notations Techie literate he didn't know what polynomials were...

[steady robin]

I'm not that dumb

[torpid ridge]

You did not, in fact, know what a polynomial was.

[native sandal]

https://tenor.com/view/monke-monkey-wth-wtf-bruh-gif-26418485

[steady robin]

But why is n²-n+41 a polynomial?

[torpid ridge]

... because it has the required form.

a_0 = 41, a_1 = -1, a_2 = 1.

[native sandal]

bro got sad

[thin berry]

you're confusing polynomial with function

[steady robin]

Oh

[smoky reef]

I don’t even think it’s a good definition for a function though

The normal one is much better