Michael Lugo
Reputation
12,596
Top tag
Next privilege 15,000 Rep.
Protect questions
 1d comment You have to estimate $\binom{63}{19}$ in $2$ minutes to save your life. Except the denominator here should be $10^{19}$. Apr 18 comment Stability of a dimer on a square grid after $n$ random steps An observation about the value of using calculations like this to make conjectures: it's easy to see that $P_1 = 3^2/2^4, P_2 = 10^2/2^8, P_3 = 35^2/2^{12}$. So one can conjecture the result from recognizing the sequence $3, 10, 35$ as ${2n + 1 \choose n}$ for $n = 1, 2, 3$. Apr 15 comment Ranking players and puzzles from performance in a single player game format This question seems appropriate to me, but it's really a statistical question. As such it's probably more appropriate for stats.stackexchange.com Apr 13 comment What percentage of prime number factorials plus 1 are themselves prime? The OEIS sequences oeis.org/A088332 and oeis.org/A002981 are relevant. The sequence looks sparse enough that I'd guess the density is 0. The paper by Caldwell and Gallot at utm.edu/staff/caldwell/preprints/primorials.pdf gives a heuristic (Conjecture 2.3) that the number of primes $n! + 1$ with $n \le N$ is asymptotic to $e^\gamma \log N$ as $N \to \infty$. Apr 12 comment How many points to prove a trigonometric identity? In this case, all the trig functions can be expressed in terms of $\exp{i \theta}$ and so both sides are rational functions of $\exp{i \theta}$; thus it should be possible to reduce this to a question about polynomials. Apr 7 comment Generating Numbers Proof I'd note that you also would need to remove trailing zeroes. For example, consider the sequence 19, 38, 76, 152, 304, 30. (You don't have to remove them right away - this could continue 30, 60, 120, 240, 24, 2 or 30, 3, 6, 12, 24, 2 - but I don't see any reason not to remove trailing zeros as they appear.) Apr 1 comment Primes Between $n$ and $2n$ For $n\ge6$ That sequence of primes is called the "Ramanujan primes": see oeis.org/A104272 Mar 30 comment Asymptotic bounds of product of $\log(i)$ It will suffice to find bounds for the log of your product, $\sum_{k=2}^n \log_2 \log_2 k$, which is done at math.stackexchange.com/questions/1084665/… Mar 23 comment Why is arccos(-1/3) the optimal angle between bonds in a methane (CH4) molecule? As is pointed out at the related question on this site math.stackexchange.com/questions/1663014/… , this is known as the Tammes problem - see e. g. en.wikipedia.org/wiki/Tammes_problem . This is not an answer but it might give you some idea where to look. Mar 21 comment What is the probability that a point chosen randomly from inside an equilateral triangle is closer to the center than to any of the edges? I like this solution because it makes it clear why the answer is a rational number. From the calculus-based answer this seems a bit mysterious. Mar 16 comment When is $\binom{2n}{n}\cdot \frac{1}{2n}$ an integer? Also see oeis.org/A067348 , which is double this sequence by definition. Mar 10 comment How to integrate when two functions are being multiplied by each other? You want "integration by parts". Mar 10 comment A five-digit number whose square has its last five digits equal to the number. That's right. I did it for $k = 2$ because doing it for $k = 1$ seemed trivial. Mar 7 comment $N=(x^2-1)(y^2-1)$ has more than one solution oeis.org/A063067 gives a list of such numbers (if you find the first three solutions 360, 504, 2304 by brute force you can find this list). That provides a link to the following notes: mathpages.com/home/kmath275.htm Feb 26 comment How to prove that if $S(n)$ is triangular number for $n \in Z_{+}$, then $a=9$? Perhaps relevant: $S(2)$ is triangular if and only if $a$ is in the sequence at oeis.org/A216134 $(1, 4, 9, 26, 55, \ldots)$. I also find the sporadic solutions $a = 2, 23$ for $n = 3$ and no others for $n \le 5, a \le 10^7$. Feb 25 comment Find two vectors in $P$ and check that their sum is not in $P$. That's right. What's more, that will hold for any plane $P$ that doesn't pass through the origin. Feb 24 comment Solve $\lim_{n\to \infty}\left(\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{n+n}\right)$ without using Riemann sums. How do you get that $\lim_{n \to \infty} H_n - \log n - \gamma = 0$ without using Riemann sums? Feb 8 comment Limit of the sequence $\lim_\limits{n\to\infty}\sin(2\pi(n^2+n^{1/2})^{1/2})$. Yes, that does it. Alternatively (Taylor seems like overkill for this to me), note that $\sqrt{1+x} < 1+x/2$, so $\sqrt{1+n^{-3/2}} - 1 < {1 \over 2} n^{-3/2}$. Feb 8 comment Limit of the sequence $\lim_\limits{n\to\infty}\sin(2\pi(n^2+n^{1/2})^{1/2})$. In this case, yes. There is no limit if $n$ is a real number but there is if $n$ is an integer. Feb 8 comment Limit of the sequence $\lim_\limits{n\to\infty}\sin(2\pi(n^2+n^{1/2})^{1/2})$. Do you mean for $n$ to be an integer or a real number?