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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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The $p$-adic valuation of powers of consecutive integers

Let $n > 0, K > 0$ integers and, for $i \in \{1,...,n\}$, let $k_i$ and $l_i$ be integers such that $k_i + l_i = K$. Assume that for some $i,j \in \{1,...,n\}$ we have $k_i \neq k_j$. Claim: ...
125 views

Squares in Lucas sequences

Good night, everyone! According to a celebrated result by J. H. Cohn, the only perfect squares in the Fibonacci sequence are $F_{0}=0$, $F_{1}=F_{2}=1$, and $F_{12}=144$. It is also known that the ...
77 views

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 $\... 0answers 168 views Conjectured primality test for numbers of the form$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^{...
355 views

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 $S_N := \sum_{t=1}^N\gamma^t e_{N-... 2answers 253 views A set, product of any two elements minus one is a perfect square The first problem of IMO 1986 asks the following: Prove that, one can find two distinct$a,b$in the set$\{2,5,13,d\}$such that$ab-1$is not a perfect square. Note that, this proves, for the ... 0answers 49 views Bounds for$\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 ... 1answer 533 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 $$1+3^a = 3^b+5^c.$$ 1answer 308 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 ... 1answer 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 ... 0answers 105 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-...
771 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 ...
201 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}{\...
920 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 ...
327 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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