# Search results for query: *

1. ### I Rewriting the Toeplitz Conjecture

The Toeplitz Conjecture (better known as the inscribed square problem) states that all Jordan curves have an inscribed square. It has been stated in the early 1900's and remains an open problem. I drew a square and then making a ton of curves that touch its four vertices: This shows that the...
2. ### I Defining the Prime Gap function

I thought I didn't put the definition for R(x) so I typed it again on accident

I saw this one on a shirt:

5. ### Dardistan, one of the world’s most multilingual places

There's a ton of "Stans" that aren't there own nations. Here are a few more: Dagestan, Tatarstan, Khusestan, Nuristan, Baluchistan, and Sistan. The region of Afghanistan, Pakistan, and Kashmir are very ethnically diverse themselves. I have a theory that this happens because of the numerous...
6. ### Did Columbus know he had discovered America?

He probably did know in his heart since he didn't find any spices or valuables like in China. But if he said that he didn't go to China and went to somewhere else, then Spain won't fund his voyages. I have also heard that he made his crew swear that one of the landmasses he visited (which we now...
7. ### I Defining the Prime Gap function

I am getting dizzy
8. ### I Defining the Prime Gap function

I'm not even in high school yet :-p
9. ### I Defining the Prime Gap function

Thanks! So $$R(x)= \min{y \in \mathbb{N}$$
10. ### I Defining the Prime Gap function

This is why PF is amazing the people keep giving out resources. TYSM!
11. ### I Defining the Prime Gap function

Thanks! So $$R(x)= \min \lbrace y \in \mathbb{N} \, : \, \pi(y) = \pi(x) + 1 \rbrace - x$$
12. ### I Defining the Prime Gap function

I don't know how to write the smallest value of ##\pi^{-1}(x)## in set notation. But I did go ahead and graph a few values of ##R(x)##: https://www.desmos.com/calculator/vacrq5jxg1
13. ### I Defining the Prime Gap function

OH. Well, the inverse prime function actually doesn't exist since ##\pi(x)## is a step function. So now what? Maybe if we think of ##\pi^{-1}## as a set of numbers and take the smallest one then we are fine?
14. ### I Defining the Prime Gap function

That's fine since we are only looking at positive integers.
15. ### I Defining the Prime Gap function

I asked Wolfram to find the inverse function of ##\frac{x}{\ln (x)}## (which is an approximation for ##\pi (x)##) and it gave me ##-xW(-\frac{1}{x})##. So an approximation for ##R(x)## is ##-(\pi(x)+1)W(-\frac{1}{\pi(x)+1}) - x##
16. ### I Defining the Prime Gap function

Thank you very much!
17. ### I Defining the Prime Gap function

No my library doesn't have it and my dad doesn't want me to buy books. Does it help?
18. ### I Defining the Prime Gap function

Hi PF! I created a function ##R(x)## that gives the gap between the largest two primes less than or equal to ##x##. To define it, I used this property: $$\pi(x+R(x))=\pi(x)+1$$ Which is true since the ##x## distance between ##\pi(x)## and ##\pi(x)+1## is ##R(x)##. If we solve for ##R(x)## we...
19. ### B Popularizing a property for n-bonacci numbers without publishing it?

Where did you get ##\varphi^3-2\varphi^2+\varphi^2-\varphi = \varphi^3-2\varphi^2+1##? I don't understand this part.
20. ### B Is this identity containing the Gaussian Integral of any use?

I couldn't edit the OP.
21. ### B Popularizing a property for n-bonacci numbers without publishing it?

Yeah I thought that since the paper would probably be too small I wouldn't be able to publish it. But maybe, who knows.

47. ### Admissions Ideas on Davidson Institute Fellows Scholarship

I found my research advisor. You can close this thread now :)
48. ### Admissions Ideas on Davidson Institute Fellows Scholarship

Does this already exist: ##K=\frac{\tan C (a^2+b^2-c^2)}{4}## I don't know tbh After thinking about it, I won't use this formula for Davidson Institute since it requires four variables Thanks for the advice!
49. ### Admissions Ideas on Davidson Institute Fellows Scholarship

I made a new formula for finding the area of oblique triangles. Would this be a good idea?
50. ### B How to merge the sum and ##x^n##?

Actually let's just stick with that. The expression is equal to (x+y)^n. Turns out that this formula is harder to use than the one we have now.