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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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        39 views

        Is the largest eigenvalue that matches an eigenvector that spans $v$ is independent of basis? [migrated]

        Let $W$ be a finate-dimensional inner product space over $\mathbb{R}$ (or $\mathbb{C}$) and let $M: W\to W$ be a self-adjoint operator. Then there exists an orthonormal basis $B=\{\phi_i\}$ consisting ...
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        72 views

        Average number of pieces of a random piecewise-linear function

        Let $I$ be a (nonempty) compact interval in $\mathbb R$ and $a_1,b_1,\ldots,a_L,b_L \in \mathbb R$. Let $\varphi$ be a piecewise function with $T \ge 2$ pieces(for example $T=2$ for the choice $\...
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        165 views

        Conjectured primality test for numbers of the form $N=4 \cdot 3^n-1$

        This is a repost of this question. Can you provide proof or counterexample for the claim given below? Inspired by Lucas-Lehmer primality test I have formulated the following claim: Let $P_m(x)=2^{...
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        2answers
        299 views

        Good upper bound for a certain sum

        Given $\gamma \in [0, 1)$, an integer $N \ge 2$, and a decreasing null sequence of positive numbers $e_1,e_2,\ldots,e_t,\ldots$, I'm interested in estimating the sum $S_N := \sum_{t=1}^N\gamma^t e_{N-...
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        2answers
        245 views

        A set, product of any two elements minus one is a perfect square

        The first problem of IMO 1986 asks the following: Prove that, one can find two distinct $a,b$ in the set $\{2,5,13,d\}$ such that $ab-1$ is not a perfect square. Note that, this proves, for the ...
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        0answers
        48 views

        Bounds for $\sum_{t=1}^Tn_t(s_t)^{-\alpha}\mu(s_t)$ where $n_t(s) = \sum_{1 \le t' \le t} 1_{\{s_{t'}=s\}}$ for $s \in [k]$ and $\mu \in \Delta_k$

        Disclaimer: I'm not certain this is the right venue for this post, but I'll give it a try... So trying prove some bounds in my ongoing work in theoretical reinforcement learning, I encountered the ...
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        502 views

        Diophantine equation $3^a+1=3^b+5^c$

        This is not a research problem, but challenging enough that I've decided to post it in here: Determine all triples $(a,b,c)$ of non-negative integers, satisfying $$ 1+3^a = 3^b+5^c. $$
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        1answer
        303 views

        What was the first elementary proof that $\pi(x)=o(x)$?

        Denote by $\pi(x)$ the number of primes $\leq x$. I'm interested in knowing who came up with the first elementary proof that $\pi(x)=o(x)$. I know that Chebyshev demonstrated elementarily before ...
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        93 views

        Formally proving that a metric is not induced by any norm in $\mathbb{R}^n$ [closed]

        What is the procedure to formally prove that no norm exists in $\mathbb{R}^n$, that induces a metric $d$? My first instinctive idea would be to show that $d$ is a metric in $\mathbb{R}^n$, but after ...
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        99 views

        How to define $``\ll"$ in higher dimension?

        Fix $C>0,$ we say $n \sim m $ if $|n-m| < C$ $ (n, m \in \mathbb Z)$ and $n\ll m$ if $n-m \leq C$ and $n\gg m$ if $n-m \geq C.$ Let $n_1, n_2, n_3, n_4 \in \mathbb Z$. Assume that $|n_1-...
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        2answers
        748 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 ...
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        198 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}{\...
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        1answer
        895 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 ...
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        1answer
        323 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 ...
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        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,...

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