# 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.'

212
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33
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3
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### A trigonometric equation: how hard could it be?

The following problem started out with a formulation in terms of complex numbers: let $\epsilon=e^{\frac{\pi i}3}$ and $z=e^{\frac{2\pi i}{3(2n-1)}}$. It's rather amusing that the following appears to ...

6
votes

0
answers

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### A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?

Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)?
Let $p$ be a prime ...

6
votes

1
answer

338
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### Integrality of a quotient of Fermat numbers

I try to prove that for every positive integers $m\ge n$, the following product is an integer:
$$\prod_{k=0}^{n-1}\frac{2^{2^m}-2^{2^k}}{2^{2^n}-2^{2^k}}.$$
But no luck.

7
votes

0
answers

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### Simple/Elementary derivation of Ramanujan's continued fraction for Hurwitz $\zeta(3,x+1)$

I came across this MSE post discussing a certain continued fraction for $\zeta(3)$ (more specifically, the Hurwitz zeta function $\zeta(s,z)$ at $s=3$) due to Ramanujan. I asked the original poster ...

2
votes

0
answers

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### How to prove this weighted sum inequality with non-increasing sequences?

Problem
I have two non-increasing sequences, $X = (x_1, x_3, x_5, \ldots, x_{n-1})$ and $Y = (y_1, y_3, y_5, \ldots, y_{n -1})$, $n$ is an even integer. I want to prove this inequality:
$$
\sum_{i=1}^{...

4
votes

0
answers

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### A combinatorial proof for equality of two $q$-series

Consider the following two $q$-series
\begin{align*}
f(q):&=\sum_{k=1}^{\infty} \frac{(-1)^{k-1}(1 + q^k)\,q^{\binom{k + 1}2}}{(1 - q^k)^2} \qquad \text{and} \\
g(q):&=\frac1{\prod_{j=1}^{\...

3
votes

1
answer

145
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### Arcular triangle inequality

Is it true that if inside a circular segment $S$, with vertices $a$ and $b$, we take two circular arcs, one from $a$ to $c$ and the other from $c$ to $b$, then the sum of the lengths of these two arcs ...

19
votes

3
answers

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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 ...

3
votes

1
answer

154
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### Embedding of half open half closed $n$-set in $n$-space

Let $n\geq 2$. Set $\Sigma= \{x\in \mathbb{R}^n: 1\leq |x|<2\}$. Assume $h:\Sigma
\rightarrow \mathbb{R}^n$ is continuous and injective.
Question: Must $h$ also be an embedding?
Some thoughts:
$h|...

0
votes

0
answers

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### Finding a point that minimizes sum of distances to a given set of lines

Given a set $L$ of size $n$ of lines in $\mathbb{R}^d$, find a point $x \in \mathbb{R}^d$ that minimizes: $$\sum\limits_{l\in L}\min\limits_{y\in l} {\lvert \lvert x-y \rvert\rvert}^2$$
I wrote a 1.5-...

1
vote

1
answer

115
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### Product/quotient of factorials beget dyadic powers

I am writing up some notes and the following occurred to me and I would like to see if there are a variety of ways to prove it. Just for reference, the identity pops out of equality between constant ...

16
votes

1
answer

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### Integral inequality: an elementary proof?

I have a very indirect proof of the following property involving a parametrized integral. If $a,a_1,\ldots,a_n\in\mathbb R^n$ (here $n\ge2$), let me denote $V(a,a_1,\ldots,a_n)$ the volume of the ...

6
votes

2
answers

419
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### Geometric proof of the three-dimensional Pythagorean theorem

All the proofs of the high-dimensional Pythagorean theorem that I know are based on induction or the additivity of the dot product. Is there any geometric construction that's similar to the well-known ...

21
votes

4
answers

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### What is the difference between elementary and non-elementary proofs of the Prime Number Theorem?

There is an easy proof of the PNT, just in a few lines, in the book by Julian Havil, "Gamma", pages 201-202. Specifically, Von Mangoldt's formula, which is very easy to derive:
$$
\psi(x) = ...

22
votes

8
answers

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### Simpler proofs using the axiom of choice [duplicate]

I am looking for examples of results which may be proven without resorting to the axiom of choice/Zorn lemma/transfinite induction but whose proof is quite simplified by the use of the axiom.
For ...

1
vote

1
answer

128
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### Analytic expression for the min value of $g(t):= \sqrt{(t-1)^2 + a^2}+ b|t|$ subject to $|t-1| \le c$

Disclaimer. Not sure this is MO-level but would really appreciate some help with this. Thanks in advance. Moved from SE.
Let $a,b,c \ge 0$, and define a function $g:\mathbb R \to \mathbb R$ by $g(t) :=...

3
votes

0
answers

104
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### Examples of "proof by generalising" [duplicate]

In a previous post I asked (Which theorems have Pythagoras' Theorem as a special case?).
Are there any compelling examples where it is significantly "easier"/"simpler" to prove ...

1
vote

1
answer

188
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### Curious identity involving the number of perfect matchings of the complete graph

Can you prove (preferably combinatorially) the following identity for the total number of perfect matchings of the complete graph $K_{2n}$, where the edges in the matching are ordered, i.e., $\binom{...

8
votes

3
answers

480
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### Equation $wxyz(w+x+y+z)=1$ in $\mathbb{Q}_+^4$

In this thread on Math.SE, Noam D. Elkies give the following parametric family of solutions in $\mathbb{Q_+}^3$ of the equation $xyz(x+y+z)=1$ (found by Euler) :
$$
x = \frac{6 t^3 (t^4-2)^2} {(4 t^4 +...

13
votes

2
answers

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### Is Gauss's generalization of Wilson's theorem non-superficially related to the classification of moduli for which primitive roots exist?

Wilson's theorem (actually proven by Lagrange) from elementary number theory states that: If $n\ge 2$ is an integer, then
$$
(n-1)! \equiv
\begin{cases}
\hfill -1 \pmod {n} &\text{ if } n \...

4
votes

1
answer

592
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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

1
answer

248
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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 ...

6
votes

1
answer

373
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### Is there any hope to prove that $g(x)>-4$ if $f(x)<0$?

I have these two functions for $x>0$, $\beta>0$ and $\alpha$ (all reals)
$$ f(x)= \frac{\alpha \; \sin (\beta \; x)}x+4 \cos (\beta\; x) ,\qquad\qquad\qquad\qquad\qquad\qquad\\
g(x)=\frac{\...

28
votes

6
answers

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### Unrigorous British mathematics prior to G.H. Hardy

I was looking at a bio-movie of Ramanujan last night. Very poignant.
Also impressed by Jeremy Irons' portrayal of G.H. Hardy.
In G.H. Hardy's wiki page, we read:
. . . "Hardy cited as his most ...

2
votes

1
answer

359
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### Prove positivity of rational functions

We say a rational function $F(z)$ is positive if the coefficients of its Maclaurin expansion, in the variable $z$, are non-negative.
In this context, let
$$F_r(z):=\frac{1 - 2z + z^r - (1 - z)^r}{(1 - ...

34
votes

17
answers

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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 ...

0
votes

0
answers

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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 ...

-1
votes

1
answer

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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 ...

0
votes

0
answers

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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 ...

2
votes

2
answers

169
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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 ...

9
votes

1
answer

293
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### Quadrisecants of knots

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 $a$, $c$, $b$, $d$, ...

4
votes

2
answers

445
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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 ...

4
votes

0
answers

126
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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 ...

2
votes

1
answer

532
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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 ...

4
votes

1
answer

193
views

### An infinite series involving Jordan's totient function

Can you provide a proof for the following claim:
$$-\displaystyle\sum_{n=1}^{\infty}\frac{J_k(n)}{n} \cdot \ln\left(1-x^n\right)=\frac{x \cdot A_{k-1}(x)}{(1-x)^k} \quad \text{for} \quad |x| < 1 \...

4
votes

1
answer

431
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### Two conjectural infinite series for $\pi$

I am looking for a proofs of the following two claims:
Claim 1.
$$\frac{2\pi}{\sqrt{3}}=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{\Omega_1(n)}}{n}$$ where $\Omega_1(n)$ is the number of prime ...

1
vote

0
answers

89
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### Pseudo-Droz-Farny circles

I would like to present a construction of 2 circles. These 2 circles are somewhat similar in appearance to the well known Droz-Farny circles that can be drawn for every isogonal conjugate pairs of ...

12
votes

2
answers

635
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### A conjectural infinite series for $\frac{\pi^2}{5\sqrt{5}}$

I am looking for a proof of the following claim:
First define the function $\chi(n)$ as follows:
$$\chi(n)=\begin{cases}1, & \text{if }n \equiv \pm 1 \pmod{10} \\
-1, & \text{if }n \equiv \pm ...

7
votes

2
answers

449
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### An infinite series involving the mod-parity of Euler's totient function

Can you prove or disprove the following claim:
First, define the function $\xi(n)$ as follows: $$\xi(n)=\begin{cases}-1, & \text{if }\varphi(n) \equiv 0 \pmod{4} \\
1, & \text{if }\varphi(n) \...

9
votes

1
answer

676
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### An infinite series involving harmonic numbers

I am looking for a proof of the following claim:
Let $H_n$ be the nth harmonic number. Then,
$$\frac{\pi^2}{12}=\ln^22+\displaystyle\sum_{n=1}^{\infty}\frac{H_n}{n(n+1) \cdot 2^n}$$
The SageMath ...

6
votes

1
answer

287
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### Lemoine-Lozada circles

I made some rookie attempt to define the 4th Lemoine circle recently. The alternative name for this circle was suggested yesterday. Further investigation revealed a family of circles associated with ...

15
votes

1
answer

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### The 4th vertex of a triangle?

I was immensely surprised and amused by the idea of the fourth side of a triangle that was introduced by B.F.Sherman in 1993. 'Sherman's Fourth Side of a Triangle' by Paul Yiu is available here. ...

6
votes

2
answers

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### The 4th Lemoine circle

The first and second Lemoine circles are well-known to geometers. According to this article the third Lemoine circle has been first discovered by Jean-Pierre Ehrmann in 2002. It is worth noting that ...

8
votes

3
answers

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### An infinite series that converges to $\frac{\sqrt{3}\pi}{24}$

Can you prove or disprove the following claim:
Claim:
$$\frac{\sqrt{3} \pi}{24}=\displaystyle\sum_{n=0}^{\infty}\frac{1}{(6n+1)(6n+5)}$$
The SageMath cell that demonstrates this claim can be found ...

13
votes

1
answer

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### Nonstandard proofs of the fundamental theorem of arithmetic

Thirty or so years ago, someone showed me a clever proof of the Fundamental Theorem of Arithmetic that did not make use of the lemma "If $p\mid ab$ then $p\mid a$ or $p\mid b$". I'm unable ...

1
vote

1
answer

100
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### Collinearity in tangential pentagon [closed]

I am looking for a proof of the following claim:
Given tangential pentagon. Touching point of the incircle and the side of the pentagon,the vertex opposite to that side and the intersection point of ...

3
votes

1
answer

155
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### The product of the lengths of two line segments that belong to Newton line [closed]

I am looking for the proof of the following claim:
Consider a family of bicentric quadrilaterals with the same inradius length and the same distance between incenter and circumcenter. Denote by $P$ ...

6
votes

1
answer

351
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### The square root of natural number expressed by an infinite series

Can you prove or disprove the following claim:
Let $U(n,P,Q)$ be the nth generalized Lucas number of the first kind and let $m$ be a natural number. Then,
$$\sqrt{m}=1+\displaystyle\sum_{n=1}^{\infty}...

4
votes

1
answer

1k
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### The constant $\pi$ expressed by an infinite series

I am looking for the proof of the following claim:
First, define the function $\operatorname{sgn_1}(n)$ as follows:
$$\operatorname{sgn_1}(n)=\begin{cases} -1 \quad \text{if } n \neq 3 \text{ and } n \...

4
votes

1
answer

553
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### The constant $e$ represented by an infinite series

In this Wikipedia article the constant $\pi$ is represented by the following infinite series: $$\pi=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}-\...