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...
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...
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...
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
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?
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##