So I was watching this video and at 1:35, I found out that: $$e^x > x^e,\quad{}x > 0$$ is a unique property of $e$. No other number does that. It seems legit, and probably is, anyway. But I find it a bit weird because $\pi$ seems to fit in place of $e$ just fine. In […]

The vertices of a uniform polyhedron all lie on a sphere. Out of curiosity, I looked at the circumradius $R$ of the $75$ polyhedra (non-prism) in the list (which assumed side $a=1$). For irrational $R$, almost all were roots of quadratics, quartics, and a few sextics that can factor over a square root: $\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7},\sqrt{11}$ (and […]

The following text appears in the Mill’s constant definition at the Wikipedia: There is no closed-form formula known for Mills’ constant, and it is not even known whether this number is rational (Finch 2003). And then refers to Finch: Library of Congress Cataloging in Publication Data , Finch, Steven R., 1959- Mathematical constants / Steven […]

I am on derivatives at the moment and I just bumped into this number $e$, “Euler’s number” . I am told that this number is special especially when I take the derivative of $e^x$ , because its slope of any point is 1. Also it is an irrational ($2.71828\ldots$) number that never ends, like $\pi$. […]

Given, $$x^n(2-x)=1\tag1$$ for $n=2,3,4,\dots$ $$(x-1)(x^2-x-1)=0\\ (x-1)(x^3-x^2-x-1)=0\\ (x-1)(x^4-x^3-x^2-x-1)=0$$ the roots of which are the golden ratio, the tribonacci constant, the tetranacci constant, and so on. Q: Is it true that for all integer $n>1$, if $y=x^{-1}$ then, $$2y\,(1-y^{n-1})(1-y^{2n+2}) = (1-y^{n+1})^3\tag2$$ and $$(1-y^{n-1})(1-y^{2n+2})^2 = (1-y^{n+1})^4\tag3$$ such that the RHS is a cube and fourth power, respectively? P.S. […]

People make such a big deal of the number $e$. I do not get why it is so important, other than the fact that $\ln(x)=\log_e(x)$. People say it is used all over mathematics and such, but they never give me examples. Where is the number $e$ used? Also, how did Euler come up with the […]

I am trying to devise a recursive prime-generating function following an intuition of a possible analogy to Mills and Wright prime-representing functions, in the present case based on logarithms. The proposed prime generating function $f(n)$ will provide not a subset but the complete set of primes being $f(1)=2$, $f(2)=3$, $f(3)=5$… and the prime-generating constant will […]

I found an interesting set of constants while exploring the properties of nonconstant functions that are invariant under the symmetry groups of regular polyhedra in the Riemann sphere, endowed with the standard chordal metric. The simplest functions I have found posessing these symmetries are rational functions, with zeroes at the vertices of a certain inscribed […]

I believe the following 4 questions I have, are all related to eachother. Question 1: Of course I’ve been using constants, variables and parameters for a long time, but I sometimes get confused with the definition. It seems to me that these terms are used very loosely. Let’s say we have a second degree polynomial […]

The closed-forms of the first three are well-known, $$x_1=\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\dots}}}}\tag1$$ $$x_2=\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{1+\sqrt[3]{1+\dots}}}}\tag2$$ $$x_3=\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\dots}}}}\tag3$$ $$x_4=\sqrt[3]{1+2\sqrt[3]{1+3\sqrt[3]{1+4\sqrt[3]{1+\dots}}}}=\;???\tag4$$ with $x_1$ the golden ratio, $x_2$ the plastic constant, and $x_3=3\,$ (by Ramanujan). Questions: Trying to generalize $x_3$, what is the value of $x_4$ to a $100$ or more decimal places? (The Inverse Symbolic Calculator may then come in handy to figure out […]

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