Michael Lugo
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 Nov 20 comment Verify the polynomial of degree $≤ 4$ is exact: $\int_0^1f(x)dx ≈ {1\over90}[7f(0) + 32f({1\over 4}) + 12f({1\over 2}) + 32f({3\over 4}) + 7f(1)]$. As a warmup, you might try to prove that $\int_0^1 f(x) \: dx = {1 \over 6} [f(0) + 4f(1/2) + f(1)]$ when $f$ is a polynomial of degree 3 or less. (This is Simpson's rule for integration.) Nov 5 comment Show that T(n)=4×T(n−1)−T(n−2) It would help to be more specific about the problem. What is $T$? Nov 4 answered Simplify $\sqrt{7+4\sqrt{3}}+\sqrt{28-10\sqrt{3}}$ Oct 23 comment What is the optimal strategy in the “Factor Game”? The fact that there's a grid doesn't matter, does it? The game plays out exactly the same on, say, a, 10-by-6 grid and a 15-by-4 grid, which both have 60 squares. Oct 21 answered Is there any density function such that the center of mass of a semicircle about its diameter is half its radius? Oct 16 answered Minimum number of fractions to be summed up to $\frac45$ Oct 14 comment A set of n integers is a complete residue modulo n if no two elements are congruent mod n. This is usually called a "complete residue system", and often it's defined to be a set of $n$ integers with no two congruent mod $n$. If you've been asked to prove this, you probably have been provided some other definition of "complete residue system". What definition do you have? Sep 28 comment How do I Approximate $\log{2}\approx 0.693$ without using the Maclaurin series? This is Feynman's approach: feynmanlectures.caltech.edu/I_22.html Sep 9 comment Probability that a natural number is a sum of two squares? Perhaps also of some interest: the probability that a natural number is a sum of three squares is 5/6. Aug 19 comment Bingo probability of a tie with 20 players It seems very unlikely to get an answer by explicitly counting, but straightforward programming to get an answer by simulation. Aug 9 awarded Good Answer Jul 29 comment Prime Numbers and a Two-Player Game $L$ is given by oeis.org/A025043 . Jul 22 comment asymptotics of Involutions recurrence relation Looks like you read that section of Knuth more carefully than me. I stand corrected. Jul 22 comment asymptotics of Involutions recurrence relation The coefficient 7/24 is given in the reference of Knuth cited by Wimp and Zeilberger (section 5.1.4 in volume 3), but the computations are quite involved. Jul 22 revised asymptotics of Involutions recurrence relation [Edit removed during grace period] Jul 21 awarded Yearling Jul 13 comment “Binomiable” numbers A list of these numbers is at oeis.org/A006987 - there doesn't seem to be much there. In particular, if there were a nice criterion I'd expect it to be listed there. But it's a fun problem - keep playing around with it! Jul 6 comment Find the parameter $m$ such that the number always be perfect square. This solution implicitly uses the fact that $x^2 + a$ is square for all integers $x$ and only if $a = 0$. This needs a proof, although it's easy: $a^2 + a$ is not square because it's between the consecutive squares $a^2$ and $(a+1)^2$. (Note this holds even if $a$ is negative.) Jun 22 comment Generating function solution to previous question $a_{n}=a_{\lfloor n/2\rfloor}+a_{\lfloor n/3 \rfloor}+a_{\lfloor n/6\rfloor}$ OEIS reference: oeis.org/A007731 Jun 10 reviewed Leave Open Reading list to master Numerical Analysis' research literature