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.'
209
questions
4
votes
1
answer
542
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 ...
7
votes
0
answers
243
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 ...
1
vote
1
answer
177
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{...
2
votes
0
answers
164
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:
$$
\sum_{i=1}^{...
4
votes
0
answers
226
views
A combinatorial proof for equality of two $q$-series
Consider the following two $q$-series
\begin{align*}
f(q):&=\sum_{k=1}^{\infty} \frac{(-1)^{k-1}(1 + q^k)\,q^{\binom{k + 1}2}}{(1 - q^k)^2} \qquad \text{and} \\
g(q):&=\frac1{\prod_{j=1}^{\...
4
votes
2
answers
503
views
Squares in Lucas sequences
Good night, everyone!
According to a celebrated result by J. H. Cohn, the only perfect squares in the Fibonacci sequence are $F_{0}=0$, $F_{1}=F_{2}=1$, and $F_{12}=144$. It is also known that the ...
3
votes
1
answer
144
views
Arcular triangle inequality
Is it true that if inside a circular segment $S$, with vertices $a$ and $b$, we take two circular arcs, one from $a$ to $c$ and the other from $c$ to $b$, then the sum of the lengths of these two arcs ...
20
votes
3
answers
3k
views
What is the simplest proof that the density of coprime pairs does not go to zero?
By density of coprime pairs, I mean the proportion of pairs integers between $1$ and $x$ which are coprime.
This is known to be asymptotically $1/\zeta(2)$.
I want something much weaker, namely that ...
3
votes
1
answer
151
views
Embedding of half open half closed $n$-set in $n$-space
Let $n\geq 2$. Set $\Sigma= \{x\in \mathbb{R}^n: 1\leq |x|<2\}$. Assume $h:\Sigma
\rightarrow \mathbb{R}^n$ is continuous and injective.
Question: Must $h$ also be an embedding?
Some thoughts:
$h|...
0
votes
0
answers
58
views
Finding a point that minimizes sum of distances to a given set of lines
Given a set $L$ of size $n$ of lines in $\mathbb{R}^d$, find a point $x \in \mathbb{R}^d$ that minimizes: $$\sum\limits_{l\in L}\min\limits_{y\in l} {\lvert \lvert x-y \rvert\rvert}^2$$
I wrote a 1.5-...
1
vote
1
answer
114
views
Product/quotient of factorials beget dyadic powers
I am writing up some notes and the following occurred to me and I would like to see if there are a variety of ways to prove it. Just for reference, the identity pops out of equality between constant ...
16
votes
1
answer
1k
views
Integral inequality: an elementary proof?
I have a very indirect proof of the following property involving a parametrized integral. If $a,a_1,\ldots,a_n\in\mathbb R^n$ (here $n\ge2$), let me denote $V(a,a_1,\ldots,a_n)$ the volume of the ...
6
votes
2
answers
407
views
Geometric proof of the three-dimensional Pythagorean theorem
All the proofs of the high-dimensional Pythagorean theorem that I know are based on induction or the additivity of the dot product. Is there any geometric construction that's similar to the well-known ...
21
votes
4
answers
3k
views
What is the difference between elementary and non-elementary proofs of the Prime Number Theorem?
There is an easy proof of the PNT, just in a few lines, in the book by Julian Havil, "Gamma", pages 201-202. Specifically, Von Mangoldt's formula, which is very easy to derive:
$$
\psi(x) = ...
13
votes
3
answers
1k
views
At what point would an elementary generalization of Bertrand's Postulate be interesting?
I know that in 1952 Jitsuro Nagura was able to show that there is always a prime between $k$ and $\frac{6k}{5}$ for $k > 24$.
At what point would an improvement on Nagura's result be interesting? ...
-3
votes
1
answer
373
views
Is it true that there are infinite palindromic primes that when squared give palindromic number? [closed]
Can you prove that there are infinite palindromic primes that when squared give a palindromic number?
22
votes
8
answers
3k
views
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
127
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) :=...
13
votes
1
answer
13k
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. ...
64
votes
6
answers
12k
views
What is the simplest proof that the density of primes goes to zero?
By density of primes, I mean the proportion of integers between $1$ and $x$ which are prime. The prime number theorem says that this is asymptotically $1/\log(x)$.
I want something much weaker, namely ...
3
votes
0
answers
102
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 ...
8
votes
3
answers
479
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) :
$$
x = \frac{6 t^3 (t^4-2)^2} {(4 t^4 +...
4
votes
2
answers
423
views
Checking elementary proofs with proof checkers
I am not sure if this is the right place to post this, but I have seen discussions related to proof checking here, so let me post it. If there is better place for it, please give me a hint as to where ...
12
votes
2
answers
2k
views
Is Gauss's generalization of Wilson's theorem non-superficially related to the classification of moduli for which primitive roots exist?
Wilson's theorem (actually proven by Lagrange) from elementary number theory states that: If $n\ge 2$ is an integer, then
$$
(n-1)! \equiv
\begin{cases}
\hfill -1 \pmod {n} &\text{ if } n \...
36
votes
1
answer
2k
views
On a remark of Tait on FLT for the exponent 3
This is one of those recreational questions that aren't really about research. I found a curious remark in an old volume of American Mathematical Monthly (1922) which I'll quote below:
In the ...
24
votes
2
answers
3k
views
A Putnam problem with a twist
This question is motivated by one of the problem set from this year's Putnam Examination. That is,
Problem. Let $S_1, S_2, \dots, S_{2^n-1}$ be the nonempty subsets of $\{1,2,\dots,n\}$ in some ...
8
votes
4
answers
2k
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 ...
1
vote
1
answer
242
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 ...
12
votes
2
answers
915
views
Intersection point of three circles
Can you provide a proof for the following proposition:
Proposition. Let $\triangle ABC$ be an arbitrary triangle with orthocenter $H$. Let $D,E,F$ be a midpoints of the $AB$,$BC$ and $AC$ , ...
2
votes
2
answers
520
views
A generalization of Napoleon's theorem
Can you provide a proof for the following proposition?
Proposition. Given an arbitrary $\triangle ABC$. The $\triangle AEB$, $\triangle BFC$ and $\triangle CDA$ are constructed on the sides of the $...
6
votes
1
answer
370
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)
$$ f(x)= \frac{\alpha \; \sin (\beta \; x)}x+4 \cos (\beta\; x) ,\qquad\qquad\qquad\qquad\qquad\qquad\\
g(x)=\frac{\...
2
votes
1
answer
462
views
Under what condition does Courant–Fischer–Weyl min-max principle hold in general?
From Wikipedia:
Let $A$ be an $n \times n$ Hermitian matrix. As with many other variational results on eigenvalues, one considers the Rayleigh–Ritz quotient $R_A :
\mathbf C^n \setminus \{0\} \to \...
28
votes
6
answers
6k
views
Unrigorous British mathematics prior to G.H. Hardy
I was looking at a bio-movie of Ramanujan last night. Very poignant.
Also impressed by Jeremy Irons' portrayal of G.H. Hardy.
In G.H. Hardy's wiki page, we read:
. . . "Hardy cited as his most ...
33
votes
16
answers
5k
views
Which theorems have Pythagoras' Theorem as a special case?
Loomis famously wrote hundreds of proofs of Pythagoras' Theorem (reference below), but these are all basically proofs "from below". Today on Twitter @panlepan mentioned Carnot's theorem ...
2
votes
1
answer
347
views
Prove positivity of rational functions
We say a rational function $F(z)$ is positive if the coefficients of its Maclaurin expansion, in the variable $z$, are non-negative.
In this context, let
$$F_r(z):=\frac{1 - 2z + z^r - (1 - z)^r}{(1 - ...
3
votes
1
answer
398
views
Theorems with many proofs
Q. What are the characteristics of theorems that seem to invite (or possess) several or even many distinct proofs?
What I have in mind are examples such as these:
Proofs that there are infinitely ...
0
votes
0
answers
88
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 ...
-1
votes
1
answer
76
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
91
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 ...
64
votes
3
answers
6k
views
Chebyshev polynomials of the first kind and primality testing
Can you provide a proof or a counterexample for the claim given below ?
Inspired by Agrawal's conjecture in this paper and by Theorem 4 in this paper I have formulated the following claim :
Let $...
2
votes
2
answers
164
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 ...
9
votes
1
answer
289
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 $a$, $c$, $b$, $d$, ...
16
votes
6
answers
2k
views
Alternative proofs sought after for a certain identity
Here is an identity for which I outlined two different arguments. I'm collecting further alternative proofs, so
QUESTION. can you provide another verification for the problem below?
Problem. Prove ...
10
votes
2
answers
901
views
Primality test for specific class of Proth numbers
Can you provide a proof or a counterexample for the following claim :
Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$
Let $N=k\cdot 2^n+1$ such ...
6
votes
4
answers
522
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 $\triangle ABD$,$\triangle ABC$,$\triangle BCD$ ...
4
votes
0
answers
123
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 ...
2
votes
1
answer
793
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 $A,B,C$ ...
1
vote
1
answer
491
views
A Zsigmondy-theorem-analogy in the generalized Collatz-problem $3x+\rho$?
Remark : I've found a rather trivial answer for this question and so very likely the premise of paralleling it with the Zsigmondy-theorem is wrong, so this question might better be retracted. I'll ...
17
votes
2
answers
994
views
Are [Wieferich] primes the only solutions to the equation $2^{k-1} \equiv 1 \pmod{k^2}$?
While studying a certain Diophantine equation in the squarefree integer $k \ge 2$, I believe I have proven the necessary restriction
$$2^{k-1} \equiv 1\!\!\pmod{k^2}. \qquad(\star)$$
Based on what ...
2
votes
1
answer
504
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 ...