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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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 $$...
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1 vote
1 answer
142 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}...
Pedja's user avatar
  • 2,673
3 votes
1 answer
92 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 ...
Pedja's user avatar
  • 2,673
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 $\triangle ABD$,$\triangle ABC$,$\triangle BCD$ ...
Pedja's user avatar
  • 2,673
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 ...
Pedja's user avatar
  • 2,673
3 votes
1 answer
111 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 ...
Pedja's user avatar
  • 2,673
2 votes
0 answers
197 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 ...
McDuck's user avatar
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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 $a(n + 1) / a (n)$? $(2.71...)$
José María Grau Ribas's user avatar
4 votes
1 answer
286 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 ...
Pedja's user avatar
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6 votes
1 answer
215 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 ...
Pedja's user avatar
  • 2,673
2 votes
1 answer
152 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 ...
Muzi's user avatar
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4 votes
1 answer
211 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 ...
Pedja's user avatar
  • 2,673
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? $$\int_0^1\frac{\log x\cdot\log(x+2)}{x+1}\,dx.$$
T. Amdeberhan's user avatar
1 vote
1 answer
171 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 ...
T. Amdeberhan's user avatar
30 votes
1 answer
1k views

Functional-analytic proof of the existence of non-symmetric random variables with vanishing odd moments

It is known that a random variable $X$ which is symmetric about $0$ (i.e $X$ and $-X$ have the same distribution) must have all its odd moments (when they exist!) equal to zero. The converse is a ...
dohmatob's user avatar
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1 vote
1 answer
297 views

A generalization of Harcourt's theorem

This question is closely related to my previous question. Can you prove the claim given below? The following claim is a conjectured generalization of Harcourt's theorem. Claim. Let $A_1,A_2 \ldots ...
Pedja's user avatar
  • 2,673
1 vote
1 answer
292 views

A formula for the area of bicentric quadrilateral

Can you provide a proof for the claim given below? The following claim is inspired by Harcourt's theorem and can be seen as its generalization to quadrilaterals. Claim. Given bicentric quadrilateral $...
Pedja's user avatar
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1 vote
1 answer
446 views

A new perspective on Lehmer's totient problem

Lehmer's totient problem asks if there are any composite integers $n$ with $\phi(n) \ | \ n-1$. It is known that any such $n$ must be odd. It must also be a charmichael number. Assume $n=4m+3$ then $\...
ASP's user avatar
  • 319
2 votes
1 answer
192 views

The centroid, the first and second Napoleon points and $X(930)$ lie on a circle

Can you provide an elementary proof for the claim given below? Preliminary definitions: $X(110)=$ focus of Kiepert parabola. $X(137)=X(110)$ of orthic triangle . $X(930)=$ anticomplement of $X(137)$ . ...
Pedja's user avatar
  • 2,673
2 votes
1 answer
173 views

Four concyclic triangle centers

Can you prove the claim given below? Inspired by Lester's theorem I have formulated the following claim: Claim. Given any scalene triangle $\triangle ABC$ . Let $D$ be the reflection of incenter in ...
Pedja's user avatar
  • 2,673
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 ...
Kim's user avatar
  • 4,034
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 $...
Pedja's user avatar
  • 2,673
2 votes
2 answers
243 views

Six concyclic points

Can you provide a proof for the following proposition: Proposition. Let $\triangle ABC$ be an arbitrary triangle with excenters $J_A$,$J_B$ and $J_C$ . Let $G$ be the orthogonal projection of the $...
Pedja's user avatar
  • 2,673
3 votes
2 answers
243 views

Four concyclic points inside bicentric quadrilateral

Can you provide a proof for the following proposition: Proposition. Let quadrilateral $ABCD$ be inscribed into a circle with center $O$ and circumscribed around a circle with center $I$. Let $X$ be a ...
Pedja's user avatar
  • 2,673
14 votes
2 answers
1k views

Euclid-style proof of Dirichlet’s theorem on primes in certain arithmetic progression

The well-known theorem of Dirichlet on primes in arithmetic progression states that given coprime natural numbers $a\le q$, there are infinitely many prime numbers congruent to $a\pmod q$. The ...
Jack L.'s user avatar
  • 1,423
14 votes
4 answers
1k views

Six points on an ellipse

Can you prove the following proposition: Proposition. Let $\triangle ABC$ be an arbitrary triangle with centroid $G$. Let $D,E,F$ be the points on the sides $AC$,$AB$ and $BC$ respectively , such ...
Pedja's user avatar
  • 2,673
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 $AC$ , ...
Pedja's user avatar
  • 2,673
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 ...
T. Amdeberhan's user avatar
-4 votes
2 answers
226 views

An elementary-looking integral inequality

This might seem a bit easy but I still like to ask it for pedagogical reasons. QUESTION. Is this inequality true for non-negative integers $n$? $$\frac{\pi}2\int_0^1x^n\sin\left(\frac{\pi}2x\right)dx\...
T. Amdeberhan's user avatar
2 votes
0 answers
145 views

Asking for a combinatorial proof of a binomial-sum

QUESTION. Is there a combinatorial proof of the below identity? $$\sum_{k=0}^{n-1}\frac{2^{2k}}{2k+1}\frac{\binom{2n}n}{\binom{2k}k}=2^{2n}-\binom{2n}n.$$ REMARK. There are many other proofs (...
T. Amdeberhan's user avatar
3 votes
2 answers
242 views

Is there a combinatorial reason for variable-independence of this binomial-coefficient identity?

Consider the following identity $$\sum_{n=0}^{R-t}\binom{n+\ell}n\binom{R-\ell-n}{R-t-n}=\binom{R+1}{t+1}.\tag1$$ It is relatively easy to give an algebraic or mechanical proof of (1). But, I like to ...
T. Amdeberhan's user avatar
0 votes
0 answers
182 views

A certain Pell Equation

Recently I came up with a positive solution $((x,y)\neq (\pm 1;0))$ to this diophantine equation $$ x^2-\left(w^2(2^{n-2}p)^2+2^n(2^{n-2}p)\right)y^2=1,\qquad n\geq 2, $$ where all variables are in $ ...
Toni Mhax's user avatar
  • 640
7 votes
2 answers
608 views

How to use the Prime Number Theorem in order to prove Selberg's Formula?

I`m reading Melvin B. Nathanson's "Elementary Methods in Number Theory" and I can't think of a way of deducing Selberg's formula (9.3) from the prime number theorem. This is one of the tasks ...
Juu's user avatar
  • 129
1 vote
1 answer
351 views

Primality test for numbers of the form $4k+3$

Can you prove or disprove the following claim: Let $n$ be a natural number of the form $4k+3$ , and let $c$ be the smallest odd prime such that $\binom{c}{n}=-1$ , where $\binom{}{}$ denotes a Jacobi ...
Pedja's user avatar
  • 2,673
1 vote
1 answer
226 views

Sign changes of a sequence

Let $f$ be an arithmetical function. Suppose that $f(n)>0$ if $n$ is in an integer set $A$ and that $f(n)<0$ for another integer set $B.$ Is there a result from number theory or an elementary ...
Khadija Mbarki's user avatar
3 votes
1 answer
150 views

Arithmetical function comparable to sine function [closed]

I was wondering if there exists or can we construct (using known arithmetic functions) an arithmetical function that has the same behaviour of the function sine or comparable to it (I mean that ...
Khadija Mbarki's user avatar
0 votes
0 answers
93 views

A primality criterion for specific class of $N=4kp^n+1$

Can you provide a proof for the following claim: Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4}\right)^m\right)$ . Let $N= 4kp^n+1 $ such that $p$ is a prime number greater ...
Pedja's user avatar
  • 2,673
10 votes
0 answers
623 views

Primality testing using Chebyshev polynomials

Can you provide a proof or a counterexample for the claim given below? Inspired by an alternative definition of the Frobenius primality test which is given in this paper I have formulated the ...
Pedja's user avatar
  • 2,673
1 vote
0 answers
93 views

Primality test for specific class of $N=12k \cdot 5^n-1$

Can you provide a proof for the following claim: Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4}\right)^m\right)$ . Let $N= 12k \cdot 5^{n} - 1 $ where $n\ge3$ , $12k <5^n$ ...
Pedja's user avatar
  • 2,673
0 votes
0 answers
121 views

Testing the primality of Mersenne and Fermat numbers using third order recurrence relation

Can you prove or disprove the claims given below? Inspired by generalization of Lucas-Lehmer test I have formulated the following claims: Claim 1 Let $M_p=2^p-1$ where $p$ is an odd prime number , ...
Pedja's user avatar
  • 2,673
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 ...
4 votes
1 answer
318 views

Primality test for $N=2^mp^n +1$

This question is related to my previous question. Can you prove or disprove the following claim: Let $N=2^mp^n+1$ , $m>0 , n>0$ and $p$ is an odd prime . If there exists an integer $a$ such ...
Pedja's user avatar
  • 2,673
7 votes
1 answer
456 views

Primality test for $N=2^a3^b+1$

Can you prove or disprove the following claim: Let $N=2^a3^b+1$ , $a>0 , b>0$ . If there exists an integer $c$ such that $$c^{(N-1)/3}-c^{(N-1)/6} \equiv -1 \pmod{N}$$ then $N$ is a prime. You ...
Pedja's user avatar
  • 2,673
0 votes
1 answer
359 views

Elementary proof for $n^2>p>n$ for all $n>1$ [duplicate]

Is there any elementary way of proving that for all natural numbers $n>1$ there exists a prime $p$ such that $n<p<n^2$. And I mean elementary, not using the Prime Number Theorem or Bertrand's ...
fisura filozofica's user avatar
2 votes
2 answers
562 views

A Pell like equation

If one takes in general $(\star)\, \,x^2-dy^2=C$ where $d$, $C$ in $\mathbb{N}$. Taking $d=w^2p^2+p$ with $w\in \mathbb{Q}\ge 1$ and $p\in \mathbb{Z}$ which is verified (explained later), for the ...
Toni Mhax's user avatar
  • 640
3 votes
0 answers
609 views

While solving the 1988 IMO problem 6, I have questions about new solutions without using Vieta Jumping [closed]

I think most of you may know the well-known problem: "Let $x$ and $y$ be positive integers such that $xy + 1$ divides $x^{2} + y^{2}$. Show that $\frac {x^{2} + y^{2}}{xy + 1}$ is the perfect ...
SG Kwon's user avatar
  • 39
6 votes
2 answers
309 views

How to understand the "boundary" of subscheme, as defined in "An elementary characterisation of Krull dimension"

In An elementary characterisation of Krull dimension and A short proof for the Krull dimension of a polynomial ring, Coquand, Lombardi, and Roy give an elementary characterization of Krull dimension, ...
Somatic Custard's user avatar
3 votes
0 answers
86 views

Hales' generalization of the stacked bases theorem (seeking a proof)

In his paper Analogues of the stacked bases theorem, published in the proceedings of a 1976 conference, A.W. Hales claimed some interesting generalizations of the stacked bases theorem for abelian ...
Jose Brox's user avatar
  • 2,962
4 votes
1 answer
179 views

Primality test for specific class of $N=8k \cdot 3^n-1$

This question is related to my previous question. Can you prove or disprove the following claim: Let $P_m(x)=2^{-m}\cdot \left(\left(x-\sqrt{x^2-4}\right)^{m}+\left(x+\sqrt{x^2-4}\right)^{m}\right)$ ...
Pedja's user avatar
  • 2,673
2 votes
1 answer
362 views

Primality test for specific class of $N=8kp^n-1$

My following question is related to my question here 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+\...
Pedja's user avatar
  • 2,673