I checked the link given to this OEIS-sequence : https://oeis.org/A081121 and apparantly the numbers $3136$ and $6789$ appear in the sequence. However, we have $$4192^2=260^3-3136$$ and $$94^2=25^3-6789$$ so the two numbers should not appear in the sequence. $1)$ Did I miss something, or is this actually an error ? $2)$ What is the proper site […]

Take $S^3$ to be the three-sphere, that is, $S^3=\lbrace (x_1,x_2,x_3,x_4):x_1^2+x_2^2+x_3^2+x_4^4=1\rbrace$. Using the stereographic projection, $S^3=\mathbb{R}^3\cup \lbrace \infty \rbrace.$ Can someone explain how the complement of the solid torus (centered at the origin) $S^1\times D^2$, where $D^2$ is a 2-disk, is also a torus? I am reading Milnor’s paper “On Manifolds Homeomorphic to the 7-Sphere,” and […]

What does one do when one needs a lot of symbols and one has exhausted the useful symbols of the latin and greek alphabets? (I say useful symbols because letters like iota (ι) and upsilon (υ) seem too close to “i” and “u” or nu (ν) to be useful.) What is the next most common […]

This question already has an answer here: Why is integration so much harder than differentiation? 6 answers

Let us say that someone was able to prove that $0=1$ using ZFC, thereby proving it inconsistent. What impact would this have on the study of meta-mathematics? Most mathematicians would just move onto a different set theory, since most mathematics is not sensitive to the exact axioms being used. Meta-mathematics, on the other hand, is. […]

Walking with my son at 3:14pm the other day, I mentioned to him, “Hey, it’s Pi Time”. My son knows 35 digits of $\pi$ (don’t ask), and knows that it’s transcendental. He replied, “is it exactly $\pi$ time?” This led to a discussion about whether there is ever a time each afternoon that is exactly […]

I have a little brother who is in high school and he just learnt the quadratic formula for finding roots of second degree polynomials. He asked me what why we learn this and how this could apply to a situation in the real world (preferably an application that would apply to his mindset). I mentioned […]

In What is mathematics? by Courant, Robbins, and Stewart, “5. An important inequality”, the authors change $n$ in this example: $$(1+p)^n\geq1+np$$ to $r$ in this example: $$(1+p)^r\geq1+rp$$ In other examples given by the book’s author, he also switches the variable. I also recall seeing something similar in some other book. Why would one do that?

How much emphasizes should be on proof on a first course in Linear Algebra? I sometimes feel that they (proofs) crowd out a coherent vision for linear algebra. However I also think a central theme of a Linear Algebra course is to learn reasoning even though it does not always succeed. The audience is first […]

Studying calculus I became aware that nonstandard analysis had some methods that that made the concept of infinitesimal concrete, so that $dx$ actually made sense. Can someone elaborate on this concept and whether there are any other things that are useful to know for a student in introductory calculus?

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