André Nicolas
Reputation
99/100 score
 Oct 23 comment What is the expected number of flips that are needed? I have given a solution here of the same problem with $5$ consecutive. In order to make the argument accessible to people not familiar with Markov chains, the solution did not mention them explicitly, though you will recognize transition probabilities. Oct 23 reviewed Leave Open Prove that $x^2 + y^2 = 3(z^2 + m^2)$ has no solutions in integer Oct 23 comment Counting permutations that respect a partial order You are welcome. The case of $2$ that you quoted is covered by a shorter version of the same argument. And the idea works in the same way for arranging $q$ types of coins. Oct 23 revised Counting permutations that respect a partial order added 420 characters in body Oct 23 answered Counting permutations that respect a partial order Oct 23 comment Proof of a sum of positive divisors @Lucian: It follows directly from the above identity and multiplicativity. Oct 23 reviewed Reopen Inequality: Find Min $S=\frac{a}{\sqrt{1-a}}+\frac{b}{\sqrt{1-b}}$ Oct 23 reviewed Leave Open How many will not be selected in repeated tries? Oct 23 answered Probability Puzzle: Exactly one of two specific balls among $N$ balls in $n$ draws. Oct 23 comment Given $0 < p < n$, prove there exists $n$ consecutive natural numbers such that each natural is divisible by at least $p$ distinct primes. We don't need the restriction, it just makes things confusing. Given any $P$, any $N$, there is a sequence of $N$ consecutives each divisible by at least $P$ primes. This was done in the solution referred to. I don't really want to write down the proof, it would be a near duplicate. Oct 23 comment Integrating $x\cdot{\cosh(x^2)}$ For $x\cosh(x^2)$, use substitution. The function $\cosh(x^2)$ of the title (but not of the question) does not have an elementary antiderivative. Oct 23 answered Question About Notation Nested Quantifiers. Oct 23 comment Given a joint PDF verify that it is a joint density function Anything other than $1$. Oct 23 comment Area and integration question, is this area under the curve? If it says the exact area between the $x$-axis and $\dots$ then I would say integrate $(x-1)(x-2)(x-3)$ from $1$ to $2$, add the integral of $-(x-1)(x-2)(x-3)$ from $2$ to $3$. Oct 23 awarded Enlightened Oct 23 comment How to seperate out a variable from a log I do not see the $\ln(1+\frac{r}{4})=\ln(\frac{5r}{4})$. Oct 23 comment How to seperate out a variable from a log We have $20\ln(1+r/4)=\ln((1+r/4)^{20})=4/3$. So $(1+r/4)^{20}=4/3$ and therefore $1+r/4=(4/3)^{1/20}$. (There are other ways.) Oct 23 awarded Nice Answer Oct 23 revised $\int_{0}^{\infty} x \cdot \cos(x^3) dx$ convergence added 116 characters in body Oct 23 answered $\int_{0}^{\infty} x \cdot \cos(x^3) dx$ convergence