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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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 ...

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