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 Jun 13 reviewed No Action Needed unit group of biquadratic fields Jun 12 reviewed No Action Needed Will relative entropy increase with majorization? Jun 11 comment integers which are sums of binomial coefficients: $\sum_i {n \choose k_i}$ Since the binomial coefficients appear in pairs $\binom{n}{k}=\binom{n}{n-k}$, your question can be reworded as asking for the number of distinct sums of the form $\sum_{k=0}^{n/2} a_k \binom{n}{k}$ with $a_k=0, 1, 2$. Thus there are at most $3^{n/2}$ possibilities. My first guess would be that maybe roughly the right order (that is, $\sqrt{3}^{n+o(n)}$ may be the right number). Certainly $M_n$ grows exponentially, as the binomial coefficients initially grow rapidly allowing for few repetitions in sums (such sets are called disassociated). Jun 11 comment In which orders can the numbers of prime factors of consecutive integers be? To add a tiny bit to Terry Tao's comment: Look for example at this paper by Rizwan Khan (cms.math.ca/cmb/v53/khanB9034.pdf ) where a problem on the simultaneous distribution of $\omega(n+i)$ is considered. Khan wants all of these to be very close to $\log \log (n+i)$ (and with some uniformity) but the method will work in your problem too. Jun 10 awarded Enlightened Jun 10 awarded Nice Answer Jun 10 revised Can we get good rational approximations in all residue classes? added 113 characters in body Jun 10 answered Can we get good rational approximations in all residue classes? Jun 9 reviewed Approve Is {6,3,7} an 'ultrahyperbolic' Coxeter group? Jun 9 awarded Nice Answer Jun 9 reviewed Leave Closed Algebraic K-theory can be seen as a generalization of Linear algebra? Jun 8 comment On the sum of consecutive primes and product of first and last @GregMartin: Surely $p^{\prime} \sim 2p \log p$? To Shivam Patel: Since you're 15 perhaps you won't be offended by this unsolicited advice (or feel free to ignore it)? I think you should take fedja's comment seriously, and use this as an opportunity to learn about the prime number theorem (for example, you can find a copy of Apostol's book on Analytic Number Theory inexpensively in India), and maybe also do a little coding so you can experiment with your questions yourself. After all, the fun in math is in learning and doing it, and not simply getting answers from others. All the best! Jun 8 comment Integral of sin(x)/sqrt(x) from 0 to \pi @fedja: Indeed! :) One final comment: Like you, I could also make the substitution $x\to x^2$, but WolframAlpha does a little bit more. It computes the integral to more than $50$ digits, and will make some attempt to find a closed form" if you type in a real number. I did try that, and it didn't turn up any simple relation to nice numbers (unsurprisingly). Jun 8 comment Integral of sin(x)/sqrt(x) from 0 to \pi @fedja: I take your point. In general I am happy to give questions the benefit of doubt. But it is also the responsibility of the asker to write a clear question. Even if the question is closed, OP could edit it and clarify why this would be research level; the help center gives directions to do so when a question is closed. So I find it difficult to feel too guilty about closing if no effort is made to improve the question. Jun 8 comment Integral of sin(x)/sqrt(x) from 0 to \pi @fedja: And where exactly does this question ask about independence from $e$ and $\pi$ and so on. Maybe there is an interesting question to be asked here, but the question does not do this. My guess based on the improper integral" in the question is that OP was perhaps worried about convergence, or how to compute it. If someone wishes to edit and make an interesting question out of this, then I'm happy to vote to reopen. Jun 8 comment Integral of sin(x)/sqrt(x) from 0 to \pi Just asking WolframAlpha to compute it, turns up the link with Fresnel integrals. Is there more to be said? Jun 8 reviewed No Action Needed Units of Endomorphism Rings of Jacobian Varieties with Real Multiplication Jun 8 reviewed Leave Open Mistakes in mathematics, false illusions about conjectures Jun 7 reviewed Reject nontrivial theorems with trivial proofs Jun 7 comment Billiard dynamics with angle of reflection a fraction of angle of incidence For any rectangular table, if the fraction is $\lambda<1$ then the angles tend to $\frac{\lambda}{\lambda+1}\frac{\pi}{2}$ and its complement alternately. This follows upon noting that $|\theta_n-\frac{\lambda}{\lambda+1}\frac{\pi}{2}|$ becomes the fraction $\lambda$ of its value at the next step. Note that the usual billiards for a rectangular table similarly conserves the first given angle and its complement.