# 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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### Fermat's last theorem for $n=7$ [migrated]

Which is the most elementary way of showing that $$x^7+y^7=z^7$$ has no positive integer solutions?
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### What is the simplest proof that the density of coprime pairs does not go to zero?

By density of coprime pairs, I mean the proportion of pairs integers between $1$ and $x$ which are coprime. This is known to be asymptotically $1/\zeta(2)$. I want something much weaker, namely that ...
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### Novel examples, proofs or results in mathematics from arithmetic billiards

The goal of the post is get a repository of mathematical results, proofs or examples by users of the site, arising from arithmetic billiards in number theory, analysis, geometry,…. Wikipedia has an ...
1 vote
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### Why can any open subset $U$ of $\mathbb{Q}^\infty$ be written as disjoint union of basic clopen subsets?

I am reading Engelen´s paper and have trouble with this proof of Lemma 2.1 (a) (link is below). It is easily seen that any non-empty open subspace $U$ of $\mathbb{Q}^\infty$ can be written as an ...
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### Which theorems have Pythagoras' Theorem as a special case?

Loomis famously wrote hundreds of proofs of Pythagoras' Theorem (reference below), but these are all basically proofs "from below". Today on Twitter @panlepan mentioned Carnot's theorem ...
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### Verification of a certain computation of VC dimension

Disclaimer: I'm not very familiar with the concept of VC dimensions and how to manipulate such objects. I'd be grateful if expects on the subject (learning theory, probability), could kindly proof ...
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### Characterization of tori/elliptic curve isogenies

I am reading Chapter 11 of Dale Husemöller's Elliptic Curves Springer book and I got stuck on Theorem (1.4) (c.f., image below). Notation and definitions: Let $L$ and $L'$ be two complex lattices ...
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### Why does this proof on the cyclicity of a prime multiplicative group not conclude that the solutions to a polynomial biject the powers of one element?

This argument comes from the first proof in Keith Conrad's collection of proofs that multiplicative groups of prime-order cyclic groups contain at least one generator. The proof asks the reader to ...
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### Analytic solution of low-dimensional Riccati equation

Consider the nonlinear map $F_i:\mathbb R^2 \to \mathbb R$ $F_i(x):=\varepsilon^2\langle x, A_i x\rangle +\varepsilon\langle b_i,x \rangle + x_i,$ where $A_i$ is some matrix and $b_i$ some vector Can ...
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Recall that a quadrisecant of a knot is a line that passes thru four points on it. If the points appear on the line in the order $a$, $b$, $c$, $d$ and on the knot in the order $a$, $c$, $b$, $d$, ...
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### Checking elementary proofs with proof checkers

I am not sure if this is the right place to post this, but I have seen discussions related to proof checking here, so let me post it. If there is better place for it, please give me a hint as to where ...
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### Factorization in the ring of integers of a particular biquadratic number field, and questions about norms

Consider the number field $K={\mathbb Q}[\sqrt{2},\sqrt{3}]$ and its ring of integers ${\mathcal O}_K$. I have been doing some calculations with this number field as a toy example, to see what can be ...
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### What is the integral representation of the exponential function $e^{1/t}$ on $(0,\infty)$?

A function $q(x)$ is said to be completely monotonic on an interval $I$ if $q(x)$ has derivatives of all orders on $I$ and $(-1)^{n}q^{(n)}(x)\ge0$ for $x\in I$ and $n\ge0$. See Chapter 1 in the ...
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Can you prove the following proposition: Proposition. Given an arbitrary triangle $\triangle ABC$. Let $D,E,F$ be the points on the sides $AB$,$BC$ and $AC$ respectively , such that $\frac{AB}{DA}=\... 2 votes 1 answer 774 views ### Three circles meet at a point [closed] I am looking for the proof of the following proposition: Proposition. Let$\triangle ABC$be an arbitrary triangle with circumcenter$O$. Let$A',B',C'$be a reflection points of the points$A,B,C$... 2 votes 0 answers 73 views ### Principal diagonals of octagon meet in a single point Can you provide a proof for the following claim: Claim. Given octagon circumscribed about an ellipse. If the vertices of the octagon lie on another ellipse then its principal diagonals meet in a ... 1 vote 0 answers 85 views ### How to estimate the highest power of 2 in the partial sum of 2-adic$\log(-1)$(i.e.$\sum_{i=1}^n\frac{2^i}{i}$)? The estimate I wanna get is $$v_2(\sum_{i=1}^n\frac{2^i}{i})\geq\min_{t\geq n+1}\{t-v_2(t)\}\tag{*}$$ where$v_2$is the 2-adic valuation, that is the highest power of 2 defined on$\mathbb{Q}$. Set$\$... 