<em id="zlul0"></em>

<dl id="zlul0"></dl>
<div id="zlul0"><tr id="zlul0"><object id="zlul0"></object></tr></div>
<em id="zlul0"></em>

<div id="zlul0"><ol id="zlul0"></ol></div>

# Questions tagged [elementary-proofs]

For questions related to 'elementary' proofs in a technical sense, which has nothing to do with the difficulty of the argument or result. A typical example would be 'elementary' proofs of the Prime Number Theorem, which avoid complex analysis. The tag is however not limited to this particular notion of 'elementary.'

99 questions
Filter by
Sorted by
Tagged with
404 views

I have discovered (and found an elementary proof of) the following $$\zeta(2k)=(-1)^{k-1}\dfrac{\pi^{2k}}{2^{4k}-2^{2k}}\left[k\dfrac{2^{2k}}{(2k)!}+\sum_{l=1}^{k-1}(-1)^{l}\dfrac{2^{2k-... 2answers 231 views ### Sum of squared nearest-neighbor distances between points in a square Let \square_2=\{(x,y): 0\leq x, y\leq1\} be the unit square in \mathbb{R}^2. Take n>1 points P_1, \dots, P_n\in\square_2. Denote the distances d_j=\min\{\Vert P_k-P_j\Vert: k\neq j\}, ... 15answers 3k views ### Geodesics on the sphere In a few days I will be giving a talk to (smart) high-school students on a topic which includes a brief overview on the notions of curvature and of gedesic lines. As an example, I will discuss flight ... 2answers 391 views ### How often does the Mertens function vanish? It is well known that the Mertens function$$M(x)=\sum _{n\leq x}\mu(n)$$has infinitely many zeros, and this seems to be a short proof. Are there known results about how often the Mertens function ... 4answers 699 views ### Different derivations of the value of \prod_{0\leq j<k<n}(\eta^k-\eta^j) Let \eta=e^{\frac{2\pi i}n}, an n-th root of unity. For pedagogical reasons and inspiration, I ask to see different proofs (be it elementary, sophisticated, theoretical, etc) for the following ... 2answers 304 views ### Prove that there exists a nonempty subset  I of  \{1,2,…,n\} such that  \sum_{i\in I}{\frac {1}{b_i}} is an integer Let  a_1,a_2,...,a_n and  b_1,b_2,...,b_n be positive integers such that any integer  x satisfies at least one congruence  x\equiv a_i\pmod {b_i} for some  i. Prove that there exists a ... 2answers 568 views ### Certain matrices of interesting determinant Let M_n be the n\times n matrix with entries$$\binom{i}{2j}+\binom{j}{2i}, \qquad \text{for $1\leq i,j\leq n$}. QUESTION. Is this true? There is some evidence. The determinant $\det(M_{2n+1})... 2answers 2k views ### A Putnam problem with a twist This question is motivated by one of the problem set from this year's Putnam Examination. That is, Problem. Let$S_1, S_2, \dots, S_{2^n-1}$be the nonempty subsets of$\{1,2,\dots,n\}$in some ... 0answers 74 views ### Relation between primeness, coprimeness, totient, and gcd function There are two number theoretic facts that seem to be unrelated at very first sight but at second sight seem to be strongly related to each other: (1) Primeness of one number and coprimeness of two ... 0answers 76 views ### Supremum of an almost surely continuous random process I was learning this proposition and now I have a question, how to prove it for an almost surely continuous process? I would be very grateful for any tips. 1answer 279 views ### A set of prime numbers Consider a non-empty set$S$of primes, with the property that, for every finite subset$S'\subset S$, all the primes dividing$\left(\prod_{k\in S'}k\right)+1$are in$S$. For instance, it can ... 1answer 77 views ### On submatrices: size bound Let$M$be a generic$2n\times 2n$matrix and fix$k\leq n$. Suppose$\mathcal{F}$is a family of submatrices under the conditions that$A\in\mathcal{F}$provided (a)$A$is a$k\times k$... 1answer 298 views ### Vandermonde determinant: modulo There is a lot of fascination with the Vandermonde determinant and for many good reasons and purposes. My current quest is more of number-theoretic. QUESTION. Let$p\equiv 3$(mod$4$) be a prime. ... 1answer 236 views ### Generating function for 3 -core partitions: Part II Let$\lambda$be an integer partition:$\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq0)$. Further, let$h_u$denote the hook-length of the cell$u$. We call$\lambda$a$t$-core partition if none of ... 1answer 177 views ### Generating function for$3$-core partitions Let$\lambda$be an integer partition:$\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq0)$. Further, let$h_u$denote the hook-length of the cell$u$. We call$\lambda$a$t\$-core partition if none of ...

15 30 50 per page
山西福彩快乐十分钟

<em id="zlul0"></em>

<dl id="zlul0"></dl>
<div id="zlul0"><tr id="zlul0"><object id="zlul0"></object></tr></div>
<em id="zlul0"></em>

<div id="zlul0"><ol id="zlul0"></ol></div>