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Mar
26
reviewed Close Determining parametric equation given 3 points
Feb
10
answered Problem 10 of Section 1.2 from Hatcher.
Dec
21
answered A little question about convergence of sequence
Dec
8
comment How did mathematicians decide on the axioms of linear algebra
"(...) the early history of vectors was complicated by people wanting to have a way to multiply vectors in analogy with multiplying real numbers or complex numbers (...)" This is very reasonable, but is there a written example?
Nov
2
awarded  Nice Question
Oct
21
revised Find the sum of all $x$ such that $y = 2 - \frac{9}{x+1}$ has an integer solution $(x, y)$
deleted 13 characters in body; edited title
Oct
16
awarded  Yearling
Oct
7
revised Finding $y$ so that $y^2 + x^2 \equiv 1 (\mod x + y)$
added 5 characters in body
Oct
7
answered Finding $y$ so that $y^2 + x^2 \equiv 1 (\mod x + y)$
Oct
1
answered Is $f$ uniformly continuous?
Sep
30
comment On the number of integers whose prime factors are congruent to 1 modulo some given number.
The last formula should be $(2n^2H_{2n}N^{(2n+1)/2n})/((2n+1) \log^2N)$, sorry. We should expect, then, $$C_m(N)\sim\frac{N}{\log^2 N}\sum_{n\geq 1} \frac{2n^2}{2n+1}H_n \frac{N^{1/2n}}{\varphi^n(m)}.$$
Sep
30
comment On the number of integers whose prime factors are congruent to 1 modulo some given number.
In general, the $n$th sum is asymptotic to $$\frac12\sum_{k=1}^n\pi(N^{k/2n})\pi(N^{(2n+1-k)/2n}),$$ which is like $$\frac12 \sum_{k=1}^n \frac{2nN^{k/2n}}{k\log(N)} \frac{N^{1-(k/2n)+(1/2n)}}{\left(1-(k/2n) +(1/2n)\right)\log(N)} =\frac{4n^2H_{2n}N^{(2n+1)/2n}}{(2n+1) \log^2N}$$ by the prime number theorem.
Sep
30
comment On the number of integers whose prime factors are congruent to 1 modulo some given number.
I think this possible in principle. Write $[1,N] = [1, N^{1/2}]\cup[N^{1/4}, N^{1/2}]\cup[N^{1/2}, N^{3/4}][N^{3/4}, N]$. Then the second sum is approximately $$\frac{1}{2}\left[\pi(N^{1/2})^2 + (\pi(N^{1/2}) - \pi(N^{1/4}))(\pi(N^{3/4}) - \pi(N^{1/2}))+ \pi(N^{1/4})(\pi(N) - \pi(N^{3/4}))\right].$$ Keeping only the relevant powers and using the prime number theorem, this should be asymptotic to $$\frac{10}{3}\frac{N^{5/4}}{\log^2(N)}.$$
Sep
30
comment On the number of integers whose prime factors are congruent to 1 modulo some given number.
Heuristically, I would expect $C_m(N)$ to be like $$\frac{1}{\varphi(m)}\sum_{p<N} 1+ \frac{1}{\varphi^2(m)}\sum_{pq<N}1+\cdots+\frac{1}{\varphi^k(m)}\sum_{pqr\cdots z < N} 1+\cdots,$$ by Dirichlet's theorem. The inner sums are probably well approximated by some manipulations involving $\pi(N^{1/k})$, which in turn might be combined with the prime number theorem to yield an asymptotic. Disclaimer: this is a stretch, I'm not familiar with such problems.
Sep
30
comment How to extend a function $f: S \rightarrow N$ to $\bar{f}: \bar{S} \rightarrow N$
The last part doesn't seem right. That it is continuous on a closed set does not suffice to claim uniform continuity.
Sep
30
answered Radius convergence of $\displaystyle\sum_{n=0}^{\infty}\dfrac{(-1)^n}{n} z^{n(n+1)}$
Sep
28
comment Tricky trig question from GRE
Heh, it is almost as if the testmaker made $\cos^2(\arccos(\pi/12)) \color{Red}{=} \cos(\pi/12)=\sqrt{\frac{1+\cos(\pi/6)}2}$...
Sep
11
comment Euler's errors?
The link is broken.
Sep
7
comment Mental Calculations
@Deepak This is very, very nice. I wonder what both the painting (photograph? Possibly religious?) and the paper in the wall mean.
Sep
1
reviewed Leave Open How many ways can a quadratic form represent a prime?