Julian Rosen
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 1h answered Prove that $2^{n-1}$ divides $\binom{n}{1} + \binom{n}{3}5 + \binom{n}{5}25 + \binom{n}{7}125 + \cdots$ for $n \geqslant 1$. 16h comment Prove that $2^{n-1}$ divides $\binom{n}{1} + \binom{n}{3}5 + \binom{n}{5}25 + \binom{n}{7}125 + \cdots$ for $n \geqslant 1$. I'm not sure I understand the constraint. This can be proved by expressing the sum as $((1+\sqrt{5})^n-(1-\sqrt{5})^n)/(2\sqrt{5})$. While this expression is similar to the one appearing in the Binet formula, the proof I have in mind doesn't use the Binet formula, or even the Fibonacci numbers. 17h comment Suppose that p,q are distinct odd primes. Suppose an integer k|pq-1 and k|lcm((p-1),(q-1)). Show that k|gcd((p-1),(q-1)). If our answers aren't identical, I think you should post your solution 18h answered Suppose that p,q are distinct odd primes. Suppose an integer k|pq-1 and k|lcm((p-1),(q-1)). Show that k|gcd((p-1),(q-1)). 1d answered Which matrices are conjugate to an integer valued matrix? Oct 5 revised An open cover characterization of connected spaces? edited body Oct 5 comment Concerning the ring of all real valued functions of bounded variation on $[a,b]$ A function of bounded variation is continuous at all but countably many points, so we can describe such a function with a countable sequence of real numbers (the value at every rational, plus the value at every point of discontinuity). This implies $B[a,b]$ has continuum cardinality. Oct 4 answered Binomial coefficient modularity conjecture Oct 4 answered Modification of Manhattan Geometry Sep 30 comment Ideals of Polynomial Rings and Field Extensions The first answer here appears to answer your question in the affirmative (though that answer has a downvote and I have not checked it). Sep 28 answered Why is there no general form for the harmonic numbers? Sep 28 comment Why is there no general form for the harmonic numbers? @AndréNicolas I know very little about expressing antiderivatives as elementary functions, but I am interested in your comment, particularly the mention of a Galois theory for such things. Can you suggest a reference? Sep 28 comment Why is there no general form for the harmonic numbers? An algebraic function $f(x)$ can be expanded as a Puiseux series in $x^{-1}$, i.e. $f(x)=\sum_{n\geq 0} a_i x^{n_i}$, with $a_i\in\mathbb{C}$, $n_i$ a sequence of rational numbers with bounded denominator heading to $-\infty$. Then $|f(x)|\sim |x|^{\alpha}$ as $x\to\infty$, where $\alpha=\min\{n_i:a_i\neq 0\}\in\mathbb{Q}$. The function $f(x)$ satisfies an identity $\sum_{i=0}^n g_i(x)f(x)^i=0$, with $g_0(x),\ldots,g_n(x)$ rational functions, and the value of $\alpha$ can be computed using the Newton polygon. Sep 28 comment Why is there no general form for the harmonic numbers? For the same reason, it isn't possible to express $H_n$ as an algebraic function of $n$. Sep 28 comment Why is there no general form for the harmonic numbers? For arbitrary $n$, this can be expressed in terms of the digamma function: $H_n=\psi(n+1)+\gamma$ Sep 28 answered Do we need finite intersections of open sets to be open? Sep 28 comment Do we need finite intersections of open sets to be open? Say $X=\{1,2,3\}$, where every subset is open except $\{1\}$. Take $f$ to be the characteristic function of $\{2\}$ and $g$ the characteristic function of $\{3\}$. Then $f$ and $g$ are continuous but $f+g$ is not. Sep 28 comment Do we need finite intersections of open sets to be open? As an example, "The sum of two real-valued continuous functions is continuous" fails for generalized topologies. Sep 25 comment Give a combinatorial proof that $\sum_{k=1}^{n} {{k} {n \choose k}^2 ={ n} {{2n-1} \choose {n-1}}}$ Do you know a combinatorial prrof of the related identity $\sum_k {n\choose k}^2={2n\choose n}$? Sep 25 comment Krull dimension $\leq$ transcendental degree There is an issue with the chain of subfields you describe: if $P_1\subset P_2\subset B$ are primes, we generally won't have $\mathrm{Frac}(B/P_1)\supset\mathrm{Frac}(B/P_2)$.