GH from MO
Reputation
36,592
115/100 score
 Apr 29 revised Find a subset such that its sum is divisible by $n$ edited tags; edited title Apr 28 revised Modified Jacobi’s theta function edited tags Apr 28 revised Is this group given presentation isomorphic to $\mathbb{Z}_2$, and why? deleted 2 characters in body Apr 26 comment Generalizing Ramanujan's “1729 story” It is classical that $r_2(k)$ and $r_3(k)$ can be arbitrary large. It is conjectured (but it is also widely open) that for $n\geq 5$ we have $r_n(k)\leq 1$. I don't know the status of $n=4$ from the top off my head. Apr 25 revised Checking Mertens and the like in less than linear time or less than $\sqrt{x}$ space edited tags Apr 25 comment Function that gives 1 only when an integer is divisible by another integer @AndrejBauer: See my "Added" section. (P.S. Trust analytic number theory.) Apr 25 revised Function that gives 1 only when an integer is divisible by another integer added 450 characters in body Apr 25 answered Function that gives 1 only when an integer is divisible by another integer Apr 24 comment Precise asymptotic of diophantine approximation @DouglasZare: We talk about different things. Of course the same limsup (or sup), when it is at most $1/3$, can be attained by continuum many (pairwise inequivalent) $\xi$'s. This harmonizes with my response. If the limsup (or sup) exceeds $1/3$, then $\xi$ is equivalent (in a certain sense) to $\sqrt{9-4m^{-2}}$ with $m$ a Markov number, and the limsup (or sup) is given in terms of this $m$, and there are countably many such $\xi$'s. Apr 24 revised On cluster points of a particular sequence edited tags Apr 24 comment Precise asymptotic of diophantine approximation @DouglasZare: Note also that by an element in the Markov spectrum I meant the quadratic irrational $\sqrt{9-4m^{-2}}$, where $m$ is a Markov number. Apr 24 comment Precise asymptotic of diophantine approximation @DouglasZare: I think no, since $\mathrm{SL}_2(\mathbb{Z})$ is countable. Two $\xi$'s are equivalent if and only if their continued fraction expansions are the same apart from finitely many coefficients and a shift of the coefficients. Apr 23 comment neat identities just for operators? This is an intense area of reseach. I think a full description of polynomial identities is known for $2\times 2$ matrices, but for higher ranks such a description is out of reach. Check out the papers quoting the last paper, and the literature on "polynomial identities for rings". Apr 23 revised neat identities just for operators? edited tags Apr 23 comment neat identities just for operators? Apr 23 comment neat identities just for operators? Apr 23 revised Precise asymptotic of diophantine approximation added 270 characters in body Apr 23 comment Precise asymptotic of diophantine approximation @Wojowu: Thanks, I will clarify my response. Apr 23 revised Precise asymptotic of diophantine approximation added 136 characters in body Apr 23 revised Precise asymptotic of diophantine approximation edited tags