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        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.'

        5
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        2answers
        601 views

        Products and sum of cubes in Fibonacci

        Consider the familiar sequence of Fibonacci numbers: $F_0=0, F_1=1, F_n=F_{n-1}+F_{n-2}$. Although it is rather easy to furnish an algebraic verification of the below identity, I wish to see a ...
        2
        votes
        0answers
        55 views

        Reference for calculating definite integral involving sines

        Recently I accidentally discovered a simple, elementary derivation of the following identity, valid for any $n,k \in \mathbb N$: \begin{align*} \frac1\pi \int_0^\pi {\rm d}x \left(\!\frac{\sin nx}{\...
        10
        votes
        1answer
        792 views

        Is there an elementary proof that there are infinitely many primes that are *not* completely split in an abelian extension?

        I'm currently in the middle of teaching the adelic algebraic proofs of global class field theory. One of the intermediate lemmas that one shows is the following: Lemma: if L/K is an abelian ...
        2
        votes
        1answer
        310 views

        Partitioning the positive integers into finitely many arithmetic progressions

        From Bóna's A Walk through Combinatorics: Prove or disprove that if we partition the positive integers into finitely many arithmetic progressions then there will be at least one arithmetic ...
        12
        votes
        3answers
        2k views

        Does anyone recognize this inequality?

        In some paper the authors make use of the following inequality without further explanation: Let $x\in\mathbb{R}^n$ with $x_1\le\cdots\le x_n$ and $\alpha\in[0,1]^n$ with $\sum_{i=1}^n \alpha_i=N\in\{1,...
        1
        vote
        1answer
        58 views

        Good UPPER bounds for $\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i$ where $(p_i)_i$ is a probability vector

        Let $x=(z_1,\ldots,z_n)$ be real vector and $(p_1,\ldots,p_n)$ be a probability vector. Question $\log(\sum_{i=1}^n p_ie^{z_i})-\sum_{i=1}^np_iz_i \le ???$ Observation This paper allows us to ...
        4
        votes
        2answers
        389 views

        Is the following recursion formula for $\zeta(2n)$ known?

        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)!}+{\displaystyle \sum_{l=1}^{k-1}(-1)^{l}\dfrac{2^{2k-...
        10
        votes
        2answers
        216 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 $d_j=\min\{\Vert P_k-P_j\Vert: k\neq j\}$, ...
        21
        votes
        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 ...
        5
        votes
        2answers
        363 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 ...
        7
        votes
        4answers
        681 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 ...
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        vote
        2answers
        286 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 ...
        9
        votes
        2answers
        550 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})...
        20
        votes
        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 ...
        0
        votes
        0answers
        71 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 ...

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