## All Questions

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### Sequences satisfying gcd(S(x), S(y)) = S(gcd(x,y))

Consider the sequence S(x) = 2^x - 1. This sequence has two interesting properties: a) If the GCD of S(x) and S(y) is S(gcd(x,y)), and b) For any prime p, S(p-1) is divisible by …
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### A question about time in Special and General Relativity.

I apologize if this question is considered too mathematically imprecise. My understanding of Special and General Relativity comes from reading books which attempt to explain them t …
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### In What Sense is Set Theory a ‘Foundation’ for Mathematics?

In what sense is set theory a foundation for mathematics? To my mind (for what that is worth), there are at least three (somewhat) distinct senses in which set 'theory' (I put "th …
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### Smoothness and curvature of geodesics in a length space

Let $X$ be a nice compact subset of $R^d$. Given a function $p: X \to R^+$, define the length of a path $\gamma \subset X$ as $\ell(\gamma) = \int_\gamma p(x) dx$, and the distance …
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### Groups with special automorphism group

I am looking for all finite groups $G$ such that for each subgroup $H$ of $G$ and each automorphism $\sigma$ of $H$ there exists an automorphism $\psi$ of $G$ whose restriction to …
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### n balls, k colors, expected color change problem

I was asked this question during my interview recently and despite the amount of thinking i put into this, I am yet to figure it out: Given $n$ balls which are painted by $k$ co …
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### Collages along composition of distributors

The construction of a collage of two categories $\bf A,B$ along a profunctor $\phi\colon \bf A\mid\hspace{-2mm}\to B$ gives a new category $\bf A \uplus_\phi B$ having as objects t …
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### translating a given boolean function to universal boolean function

A Boolean function U($z_1$, $z_2$ ..... , $z_m$) is universal for given n > 1 if it realizes all Boolean functions f($x_l$ ..... $x_n$) by substituting for each $z_i$ with a variab …
In "On computing factors of cyclotomic polynomials", Richard P. Brent gives the identity $$4 \Phi_n(x) = A_n(x)^2 - (-1)^{(n-1)/2} n B_n(x)^2 \qquad (1)$$ where $n$ is odd squa …