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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.'

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        4
        votes
        2answers
        404 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
        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 $d_j=\min\{\Vert P_k-P_j\Vert: k\neq j\}$, ...
        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 ...
        6
        votes
        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 ...
        7
        votes
        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 ...
        2
        votes
        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 ...
        9
        votes
        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})...
        21
        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
        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 ...
        1
        vote
        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.
        10
        votes
        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 ...
        2
        votes
        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 $k\times k$ ...
        2
        votes
        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. ...
        3
        votes
        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 ...
        8
        votes
        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 ...

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