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.'
215 questions
1
vote
0
answers
127
views
Some property of the greatest prime factor
Let $n$ be a positive integer $\geq 2$ et denote by $ P^{+}(n)$ the greatest prime factor of $n$ my question is as follows:
If $a$ and $b$ are two numbers, is there any method to express or to bound $...
- 1,257
6
votes
1
answer
743
views
Which part(s) of this proof of Goodstein's Theorem are not expressible in Peano arithmetic?
EDIT: Noah Schweber helpfully points out that $\mathsf{ACA}_0$ is a conservative extension of Peano arithmetic in which certain aspects of my proof not expressible in Peano arithmetic are expressible. ...
- 708
3
votes
0
answers
1k
views
Formula for $\pi$ involving exponents of Mersenne primes
Can someone provide a proof for the following claim?
$$\pi=\dfrac{S_0S_2}{M_3M_5} \cdot\left(\displaystyle\prod_{p \equiv 1 \pmod{4} } \frac{p}{p-1}\right) \cdot \left(\displaystyle\prod_{p \equiv 3 \...
- 2,661
34
votes
3
answers
5k
views
A trigonometric equation: how hard could it be?
The following problem started out with a formulation in terms of complex numbers: let $\epsilon=e^{\frac{\pi i}3}$ and $z=e^{\frac{2\pi i}{3(2n-1)}}$. It's rather amusing that the following appears to ...
- 43.2k
6
votes
0
answers
381
views
A possible variant of Zagier's one-sentence proof for Fermat's sum of two squares theorem?
Is it possible to modify Zagier's one-sentence proof of Fermat's sum of two squares theorem (see here) to prove certain non-trivial cases of Jacobi's four-square theorem (see here)?
Let $p$ be a prime ...
- 81
6
votes
1
answer
346
views
Integrality of a quotient of Fermat numbers
I try to prove that for every positive integers $m\ge n$, the following product is an integer:
$$\prod_{k=0}^{n-1}\frac{2^{2^m}-2^{2^k}}{2^{2^n}-2^{2^k}}.$$
But no luck.
- 3,998
7
votes
0
answers
275
views
Simple/Elementary derivation of Ramanujan's continued fraction for Hurwitz $\zeta(3,x+1)$
I came across this MSE post discussing a certain continued fraction for $\zeta(3)$ (more specifically, the Hurwitz zeta function $\zeta(s,z)$ at $s=3$) due to Ramanujan. I asked the original poster ...
- 833
2
votes
0
answers
181
views
How to prove this weighted sum inequality with non-increasing sequences?
Problem
I have two non-increasing sequences, $X = (x_1, x_3, x_5, \ldots, x_{n-1})$ and $Y = (y_1, y_3, y_5, \ldots, y_{n -1})$, $n$ is an even integer. I want to prove this inequality:
$$
\sum_{i=1}^{...
- 29
4
votes
0
answers
235
views
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}^{\...
- 43.2k
3
votes
1
answer
147
views
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 ...
- 19.1k
20
votes
3
answers
3k
views
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 ...
- 19.1k
3
votes
1
answer
157
views
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
92
views
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-...
- 19
1
vote
1
answer
118
views
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 ...
- 43.2k
16
votes
1
answer
1k
views
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 ...
- 52.4k
6
votes
2
answers
430
views
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 ...
- 1,431
21
votes
4
answers
4k
views
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:
$$
\psi(x) = ...
- 221
23
votes
8
answers
3k
views
Simpler proofs using the axiom of choice
I am looking for examples of results which may be proven without resorting to the axiom of choice/Zorn lemma/transfinite induction but whose proof is quite simplified by the use of the axiom.
For ...
1
vote
1
answer
130
views
Analytic expression for the min value of $g(t):= \sqrt{(t-1)^2 + a^2}+ b|t|$ subject to $|t-1| \le c$
Disclaimer. Not sure this is MO-level but would really appreciate some help with this. Thanks in advance. Moved from SE.
Let $a,b,c \ge 0$, and define a function $g:\mathbb R \to \mathbb R$ by $g(t) :=...
- 6,853
3
votes
0
answers
106
views
Examples of "proof by generalising" [duplicate]
In a previous post I asked (Which theorems have Pythagoras' Theorem as a special case?).
Are there any compelling examples where it is significantly "easier"/"simpler" to prove ...
- 346
1
vote
1
answer
193
views
Curious identity involving the number of perfect matchings of the complete graph
Can you prove (preferably combinatorially) the following identity for the total number of perfect matchings of the complete graph $K_{2n}$, where the edges in the matching are ordered, i.e., $\binom{...
- 109
8
votes
3
answers
488
views
Equation $wxyz(w+x+y+z)=1$ in $\mathbb{Q}_+^4$
In this thread on Math.SE, Noam D. Elkies give the following parametric family of solutions in $\mathbb{Q_+}^3$ of the equation $xyz(x+y+z)=1$ (found by Euler) :
$$
x = \frac{6 t^3 (t^4-2)^2} {(4 t^4 +...
- 655
13
votes
2
answers
2k
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
$$
(n-1)! \equiv
\begin{cases}
\hfill -1 \pmod {n} &\text{ if } n \...
- 2,075
4
votes
1
answer
645
views
Novel examples, proofs or results in mathematics from arithmetic billiards
The goal of the post is get a repository of mathematical results, proofs or examples by users of the site, arising from arithmetic billiards in number theory, analysis, geometry,….
Wikipedia has an ...
1
vote
1
answer
258
views
Why can any open subset $U$ of $\mathbb{Q}^\infty$ be written as disjoint union of basic clopen subsets?
I am reading Engelen´s paper and have trouble with this proof of Lemma 2.1 (a) (link is below).
It is easily seen that any non-empty open subspace $U$ of $\mathbb{Q}^\infty$ can be
written as an ...
- 113
6
votes
1
answer
376
views
Is there any hope to prove that $g(x)>-4$ if $f(x)<0$?
I have these two functions for $x>0$, $\beta>0$ and $\alpha$ (all reals)
$$ f(x)= \frac{\alpha \; \sin (\beta \; x)}x+4 \cos (\beta\; x) ,\qquad\qquad\qquad\qquad\qquad\qquad\\
g(x)=\frac{\...
user329770
29
votes
6
answers
6k
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 ...
- 439
2
votes
1
answer
378
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 - ...
- 43.2k
35
votes
17
answers
6k
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
97
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 ...
- 6,853
-1
votes
1
answer
91
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 ...
- 231
0
votes
0
answers
107
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 ...
- 101
2
votes
2
answers
173
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
303
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$, ...
- 45k
4
votes
2
answers
584
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 ...
- 1,018
4
votes
0
answers
140
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 ...
- 25.8k
2
votes
1
answer
563
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 ...
- 1,101
4
votes
1
answer
205
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 \...
- 2,661
4
votes
1
answer
441
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 ...
- 2,661
1
vote
0
answers
97
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 ...
- 445
12
votes
2
answers
664
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:
$$\chi(n)=\begin{cases}1, & \text{if }n \equiv \pm 1 \pmod{10} \\
-1, & \text{if }n \equiv \pm ...
- 2,661
7
votes
2
answers
458
views
An infinite series involving the mod-parity of Euler's totient function
Can you prove or disprove the following claim:
First, define the function $\xi(n)$ as follows: $$\xi(n)=\begin{cases}-1, & \text{if }\varphi(n) \equiv 0 \pmod{4} \\
1, & \text{if }\varphi(n) \...
- 2,661
9
votes
1
answer
692
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 ...
- 2,661
6
votes
1
answer
311
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 ...
- 445
15
votes
1
answer
17k
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. ...
- 445
6
votes
2
answers
505
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 ...
- 445
8
votes
3
answers
1k
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 ...
- 2,661
14
votes
1
answer
1k
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 ...
- 19.7k
1
vote
1
answer
111
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 ...
- 2,661
3
votes
1
answer
158
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 $P$ ...
- 2,661