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.'

Filter by
Sorted by
Tagged with
8 votes
3 answers
451 views

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 +...
  • 583
11 votes
2 answers
1k views

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 \...
  • 1,684
2 votes
1 answer
275 views

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
132 views

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
353 views

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{\...
  • 61
26 votes
6 answers
5k views

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 ...
  • 371
2 votes
1 answer
291 views

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 - ...
33 votes
16 answers
5k views

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
71 views

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 ...
  • 5,870
-1 votes
1 answer
56 views

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
59 views

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
135 views

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 ...
  • 192
9 votes
1 answer
257 views

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
333 views

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 ...
  • 898
4 votes
0 answers
97 views

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 ...
  • 24.1k
1 vote
1 answer
320 views

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 ...
  • 440
4 votes
1 answer
162 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 \...
  • 2,645
4 votes
1 answer
395 views

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 ...
  • 2,645
1 vote
0 answers
78 views

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
602 views

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 ...
  • 2,645
6 votes
2 answers
415 views

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) \...
  • 2,645
9 votes
1 answer
614 views

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 ...
  • 2,645
5 votes
1 answer
235 views

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 ...
3 votes
0 answers
185 views

Four infinite series involving Riemann zeta function

Can you provide a proof for at least one of the claims given below? It is known that $\pi=\displaystyle\sum_{n=1}^{\infty}\frac{3^n-1}{4^n} \cdot \zeta(n+1)$ where $\zeta$ denotes Riemann zeta ...
  • 2,645
12 votes
1 answer
8k views

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
373 views

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
925 views

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 ...
  • 2,645
13 votes
1 answer
983 views

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 ...
  • 17.6k
1 vote
1 answer
65 views

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 ...
  • 2,645
3 votes
1 answer
135 views

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 $P$ ...
  • 2,645
6 votes
1 answer
277 views

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}...
  • 2,645
4 votes
1 answer
475 views

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 \...
  • 2,645
4 votes
1 answer
380 views

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}-\...
  • 2,645
4 votes
2 answers
186 views

Six conelliptic points

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}=\...
  • 2,645
2 votes
1 answer
738 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 $A,B,C$ ...
  • 2,645
2 votes
0 answers
63 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 ...
  • 2,645
1 vote
0 answers
76 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 $$...
user avatar
1 vote
1 answer
119 views

Generalizing Bottema's theorem

Can you provide another proof for the claim given below? Claim. In any triangle $\triangle ABC$ construct triangles $\triangle ACE$ and $\triangle BDC$ on sides $AC$ and $BC$ such that $\frac{AE}{AC}...
  • 2,645
3 votes
1 answer
90 views

Equal sums of line segments

I would like to see a proof of the following Claim. Let $A_1,A_2,A_3,A_4,A_5$ be vertices of bicentric pentagon. Let $B_1$ be the intersection point of $A_1A_3$ and $A_2A_5$, $B_2$ the intersection ...
  • 2,645
6 votes
4 answers
423 views

Necessary and sufficient condition for quadrilateral to be cyclic

Can you provide a proof for the following proposition: Proposition. Given any quadrilateral $ABCD$. Let $P,Q,R,S$ be nine-point centers of triangles $\triangle ABD$,$\triangle ABC$,$\triangle BCD$ ...
  • 2,645
8 votes
4 answers
1k views

Three circles intersecting at one point

Can you provide a proof for the following proposition: Proposition. Let $\triangle ABC$ be an arbitrary triangle with nine-point center $N$ and circumcenter $O$. Let $A',B',C'$ be a reflection points ...
  • 2,645
2 votes
1 answer
76 views

Collinearity of three significant points of bicentric pentagon

Can you provide a proof for the following claim? Claim. Given bicentric pentagon. Consider the triangle whose sides are two diagonals drawn from the same vertex and side of pentagon opposite from ...
  • 2,645
2 votes
0 answers
123 views

Polyhedron - sphere intersection

newbie here. I'd like to ask you, if you know some brief, but somewhat solid proof of a convex polyhedron and a sphere centered at one of its vertices (with small enough radius, so it intersects only ...
  • 39
11 votes
3 answers
1k views

What is the limit of $a (n + 1) / a (n)$?

Let $a(n) = f(n,n)$ where $f(m,n) = 1$ if $m < 2 $ or $ n < 2$ and $f(m,n) = f(m-1,n-1) + f(m-1,n-2) + 2 f(m-2,n-1)$ otherwise. What is the limit of $a(n + 1) / a (n)$? $(2.71...)$
4 votes
1 answer
193 views

Collinearity in bicentric polygons

Can you provide a proofs for the following two claims? Claim 1. The circumcenter, the incenter, and the intersection of the principal diagonals in a bicentric even-sided polygon are collinear. Claim ...
  • 2,645
6 votes
1 answer
189 views

Necessary and sufficient condition for tangential polygon to be cyclic

Can you prove or disprove the following claim? Claim. Let $A_1,A_2, \ldots ,A_n$ be the vertices of an $n$-sided tangential polygon and let $B_1,B_2, \ldots ,B_n$ be the contact points of the ...
  • 2,645
2 votes
1 answer
135 views

Stability estimates on quotients of the form $ \frac{\prod_{j=1}^n a_j}{\prod_{j=1}^n b_j} $

Suppose that $a_j,b_j \in \mathbb C$ are complex numbers, $j=1,\dots,n$, with the property that $|a_j|,|b_j| \geq c > d >0$ where $c,d$ are positive real numbers. I'm interested in the stability ...
  • 133
4 votes
1 answer
200 views

Point of concurrency [closed]

I am looking for the proof of the following claim: Claim: Let $\triangle ABC$ be an arbitrary triangle, $D$ its nine-point center and $E,F,G$ are the nine-point centers of the triangles $\triangle ...
  • 2,645
9 votes
2 answers
1k views

A tricky integral to evaluate

I came across this integral in some work. So, I would like to ask: QUESTION. Can you evaluate this integral with proofs? $$\int_0^1\frac{\log x\cdot\log(x+2)}{x+1}\,dx.$$
1 vote
1 answer
139 views

A binomial convolution of Catalan numbers vs "utterly odd numbers"

An integer is called utterly odd if the terminal string of $1$’s in its binary representation has odd length. A number $2^{k+1}m+(2^k-1)$ where $m\geq0$ (every non-negative integer has this form) is ...