Here, a lower and upper bound for the $n$-th prime are given. Applying the given bounds $$n(\ln(n\cdot\ln(n))-1)<p_n<n\cdot\ln(n\cdot\ln(n))$$ and the approximation $$p_n\approx n(\ln(n\cdot\ln(n))-1)+\frac{n(\ln(\ln(n))-2)}{\ln(n)}$$ we get that $p_{10^{100}}$ is somewhere between $2.346977\cdot 10^{102}$ and $2.35698\cdot 10^{102}$ and approximately $2.3471\cdot 10^{102}$ , so it has $103$ digits. How many digits can we determine of the googol-th prime with […]

I want to ask if there is some book that treats Differential Geometry without assuming that the reader knows General Topology. Well, many would say: “oh, but what’s the problem ? First learn General Topology, and you’ll understand Differential Geometry even better!” and I agree with that, but my point is: I’m a student of […]

This is more a question on History than proof itself. About a decade ago, a college professor and a Math coach told us about this beautiful theorem: Every multiple of 6 can be written as a sum of four cubes The proof of the theorem is elementary as well as elegant. Consider $(n+1)^3 + (n-1)^3 […]

Background: It’s straightforward to check that the average (i.e. the mean) of the roots of a nonlinear polynomial equals the average of the roots of its derivative: if $$f(x) = x^n + a_{n-1} x^{n-1} + \cdots a_0$$ then the roots of $f(x)$, counting multiplicities, sum to $- a_{n-1}$, while the roots of $$f'(x) = n […]

I am looking for detailed references containing proofs of inclusion relationships between different $L^p$ spaces and multiple counterexamples of functions in one but not the others.

Suppose that $p_1:E_1\to B$, $p_2:E_2 \to B$ are two $C^\infty$ fiber bundles which are $C^1$ isomorphic. That is, there exists a $C^1$ diffeomorphism $f:E_1\to E_2$ satisfying $p_2 \circ f = p_1$. Question: Does it follow that $p_1:E_1\to B$ and $p_2:E_2 \to B$ are $C^\infty$ isomorphic? Motivation: If $M, N$ are two $C^\infty$ manifolds which are […]

I’ve worked through a computation-heavy, “standard” but quite nonrigorous treatment of multivariable calculus in the past. What book would do well as a rigorous (but not overly) “second course”? In particular, I’m looking for a book that treats differential forms treats the inverse and implicit function theorems and leads well into an intro manifolds book […]

I recently placed a question based on quadratics and received a few valuable answers. One of them was a comment in an answer with a link in it which I found useful. But unfortunately the webpage (of which the link was sent) was in Russian (which is totally a foreign language to me) and so […]

I have been doing a project about $\operatorname{Ext}^n$ functors for my commutative algebra class. I used the approach via extensions of degree n. Basically I have shown the long exact sequence associated to a short exact sequence for $\operatorname{Hom}_{\mathcal{A}}$ where $\mathcal{A}$ is an abelian category. To apply this construction I have proven the Chinese remainder […]

As the title says, I’m wondering whether there is any known closed-from for the following series: $$\sum_{k=0}^{\infty} \frac{1}{(k!)!}$$ Here I don’t mean the double factorial (treated here) when I’m writing $(k!)!$, but the factorial of the integer $k!$. Trying on WolframAlpha, I get the funny value $2.501388888888888888888890500626459…$, but no closed-form. If you have any reference […]

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