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 ...

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54 views

### On submatrices: size bound

Let $M$ be a generic $2n\times 2n$ matrix and fix $k\leq n$.
Suppose $\mathcal{F}$ is a family of submatrices under the conditions that $A\in\mathcal{F}$ provided
(a) $A$ is a $k\times k$ ...

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votes

**1**answer

243 views

### Vandermonde determinant: modulo

There is a lot of fascination with the Vandermonde determinant and for many good reasons and purposes. My current quest is more of number-theoretic.
QUESTION. Let $p\equiv 3$ (mod $4$) be a prime. ...

**3**

votes

**1**answer

186 views

### Generating function for 3 -core partitions: Part II

Let $\lambda$ be an integer partition: $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq0)$. Further, let $h_u$ denote the hook-length of the cell $u$.
We call $\lambda$ a $t$-core partition if none of ...

**4**

votes

**1**answer

106 views

### Generating function for $3$-core partitions

Let $\lambda$ be an integer partition: $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq0)$. Further, let $h_u$ denote the hook-length of the cell $u$.
We call $\lambda$ a $t$-core partition if none of ...

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votes

**2**answers

165 views

### A link between hooks and contents: Part II

This is a question in the spirit of an earlier problem.
Let $\lambda$ be an integer partition: $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq0)$.
Recall also the notation for the content of a cell $...

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**2**answers

120 views

### A link between hooks, contents and parts of a partition

Let $\lambda$ be an integer partition: $\lambda=(\lambda_1\geq\lambda_2\geq\dots\geq0)$. Denote its conjugate partition by $\lambda'$. For example, if $\lambda=(4,3,1)$ then $\lambda'=(3,2,2,1)$.
...

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**0**answers

187 views

### Why are the medians of a triangle concurrent? In absolute geometry

This fact holds true in absolute geometry, and I would like to see an elementary synthetic proof not using the classification of absolute planes (Euclidean and hyperbolic planes) and specific models. ...

**21**

votes

**1**answer

834 views

### A proof required for this identity [duplicate]

Experiments support the below identity.
Question. Is this true? Combinatorial proof preferred if possible.
$$\sum_{m=0}^n\binom{n-\frac13}m\binom{n+\frac13}{n-m}(1+6m-3n)^{2n+1}
=\left(\frac43\...

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votes

**0**answers

233 views

### Conjectured primality tests for specific classes of $k\cdot b^n \pm 1$

Can you provide proofs or counterexamples for the claims given below?
Inspired by Lucas-Lehmer-Riesel primality test I have formulated the following two claims:
First claim
Let $P_m(x)=2^{-m}\...

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vote

**1**answer

309 views

### Primality test for generalized Fermat numbers

This question is successor of Primality test for specific class of generalized Fermat numbers .
Can you provide a proof or a counterexample for the claim given below?
Inspired by Lucas–Lehmer–Riesel ...

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votes

**1**answer

146 views

### $2$-adic valuations and sum of divisor function

Consider the sum of $k^{th}$-power of divisors of $n$, denoted
$$\sigma_k(n)=\sum_{d\vert n}d^k.$$
Let $\nu_p(x)$ stand for the $p$-adic valuation of the integer $x$.
The following appears to be ...

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votes

**1**answer

567 views

### Roots and relation between polynomials and their derivatives

This is probably easy but it might be interesting. Here goes $\dots$
Let $P\in\mathbb{R}[x]$ be a polynomial of degree $n>2$ and $P'=\frac{dP}{dx}$. If $x_1, x_2, \dots, x_n$ are the roots of $P(x)...

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votes

**1**answer

271 views

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

This question is successor of Compositeness test for specific class of $N=k \cdot b^n-1$ .
Can you provide a proof or a counterexample to the following claim :
Let $P_m(x)=2^{-m}\cdot \left(\left(...

**1**

vote

**0**answers

128 views

### Algebraic equivalence of cycles and Chow varieties

Let $p,d\geq 0$ be two integers and let $X\subseteq\mathbb{P}^N$ be acomplex projective variety. Denote the Chow variety of $X$ consisting of $p$-cycles of degree $d$ by $\mathcal{C}_{p,d}(X)$. I'm ...

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**2**answers

397 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 $N=k\cdot 2^n+1$...

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**1**answer

128 views

### How to choose compactly supported smooth $h$ so $h^2(x)+ h^2(x-1)=1$ for all $x\in [0,1],$ and $\int_{-3/4}^{3/4} |h(x)|^2 dx =3/2$? [closed]

It is known that we may choose smooth $f:\mathbb R \to [0,1]$ such that $f(x)=1$ if $x\geq \frac{3}{4} $ and $f(x)=0$ if $x\leq -\frac{3}{4}+1.$
Define $h(x)= \sin (\frac{\pi}{2} f(x+1))$ if $x\...

**-3**

votes

**1**answer

218 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?

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**1**answer

227 views

### Simplifying a double sum of inverses

Let $$f(n) = 2 \sum\limits_{a = 2}^{n - 1} \sum\limits_{b = n + 1}^{n + a - 1}\frac{1}{ab} .$$
One can see that $$\lim\limits_{n \to \infty}f(n) = 2\int\limits_0^1 \frac{dx}{x} \int\limits_1^{1 + x}\...

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votes

**1**answer

409 views

### Prove that these are polynomials

Define the functions
$$F_n(q)=\frac1{(1-q)^{2n}}\sum_{k=0}^n(-q)^k\frac{2k+1}{n+k+1}\binom{2n}{n-k}
\prod_{j=0,\,j\neq k}^n\frac{1+q^{2j+1}}{1+q}.$$
The numbers $\frac{2k+1}{n+k+1}\binom{2n}{n-k}$ ...

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votes

**1**answer

944 views

### What is the value of this double sum in closed form?

I encountered the following double sum which requires an evaluation.
Is there a closed form for this?
$$\sum_{n=0}^{\infty}\frac{\sum_{k=0}^n\binom{n}k^{-1}}{(n+1)(n+2)}.$$
Incidentally, it ...

**2**

votes

**0**answers

227 views

### Conjectured initial values of Inkeri's primality test for Fermat numbers

This is a repost of this question .
Can you provide a proof or a counterexample to the claim given below ?
First , we shall give a definition of the Inkeri's primality test for Fermat numbers :
...

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vote

**1**answer

185 views

### Another question on strengthening the Sylvester-Schur Theorem

I've found an argument that if valid suggests that there is always a prime $p > n$ that divides ${{x+n} \choose n}$ when $x > \pi(n)$.
As a math amateur, I am always doubtful about my results ...

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votes

**2**answers

402 views

### These polynomials are always either even or odd [duplicate]

The falling factorials are $(x)_n=x(x-1)\cdots(x-n+1)$ with $(x)_0:=1$. Define the forward shift $E$ and the discrete derivative $\delta=E-1$, respectively, by
$$Ef(x)=f(x+1) \qquad \text{and} \qquad \...

**1**

vote

**0**answers

140 views

### Combination of $k$-powers and divisibility [closed]

Let $n$ be a positive integer. Determine integers, $n+1\leq r\leq 3n+2$, such that for all integers, $a_1,a_2,\dots,a_m$, $b_1,b_2,\dots,b_m$, satisfying the equations
$$
a_1b_1^k+a_2b_2^k+\cdots+...

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**0**answers

61 views

### Can this function satisfy Song conditions?

Let $r_{s,k}(n)$ be the number of representation of a naturel number $l$ as a sum of $s$ positive $k$-th powers.
Joung Min Song introduced some conditions to study asymtotic behavior of some positive ...

**1**

vote

**1**answer

176 views

### An elementary sequence question [closed]

Below is a problem, from an old Silk Road olympiad.
Define an infinite sequence, $a(n)$, such that, $a(1)=a(2)=1$;
$$
a(n)=a(a(n-1))+a(n-a(n-1)),\forall n\geq 3.
$$
Show that, for every $n\geq 1$, $a(...

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votes

**1**answer

212 views

### sequence generated with polynomials

Below is an old olympiad problem, that turned out to be notoriously hard, that we couldn't solve it. If anyone has a solution for it, I'd be grateful.
Let $P(x)=x+1$, and $Q(x)=x^2+1$. We consider ...

**48**

votes

**2**answers

3k 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 $...

**3**

votes

**1**answer

186 views

### On decompositions of integers as a linear combination of $(1, 2, 3,\ldots)$

Edited: Given integer $N\geq 0$, let $$I(N):=\Bigl\{(n_k)_{k\geq 1}\in {\mathbb N}^\infty \,:\, n_k\geq 0, \sum_{k\geq 1}kn_k = N \Bigr\}$$ be the set of all decompositions of $N$ as a linear ...

**5**

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**1**answer

244 views

### An elementary question about a sequence of numbers

Let $\lambda_n$ be an increasing and unbounded sequence of positive real numbers and $a_n$ be a sequence of real numbers such that
$$\sum_{n=1}^\infty a_n \lambda_n^k=0 \ \ \text{ for all }\ \ k\geq ...

**0**

votes

**1**answer

285 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 ...

**1**

vote

**1**answer

49 views

### A problem with elementary inequality involving probabilities and Brier scoring rule

I am trying to prove certain relations between certain values of the so called Brier inaccuracy measure (Brier scoring rule).
Given a vector $p = (p_1, \ldots p_n)$, where $p_1 + \ldots p_n = 1$ and $...

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**0**answers

553 views

### An elementary proof that, for every fixed $n \in \mathbf N^+$, there are infinitely many primes $\equiv -1 \bmod n$

This morning, I made a comment to a comment to a question of Ayman Moussa, only to point out that, among many others, there is an elementary proof of Dirichlet's theorem on the existence of infinitely ...

**4**

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**3**answers

874 views

### Solution to a Diophantine equation

Find all the non-trivial integer solutions to the equation
$$\frac{a}{b+c}+\frac{b}{a+c}+\frac{c}{a+b}=4.$$

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**2**answers

607 views

### Are simplified elementary proofs if valid interesting to the professional mathematical community [closed]

For the last 10+ years, as a math amateur, I worked nightly on understanding the distribution of primes and the classic results in the history of Fermat's Last Theorem.
I have made numerous mistakes ...

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**1**answer

565 views

### Going beyond the Sylvester and Schur theorem with regard to $x,x+1,\dots,x+n-1$

I was recent reading through Paul Erdos's classic elementary proof of Sylvester-Schur. It occurred me that there is a simple argument that when $x$ is sufficiently large and if $p_i$ represents the $...

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votes

**2**answers

255 views

### complex polynomials and inequalities

Let $z_1,z_2,\dots,z_n\in\Bbb{C}$ be distinct and $w_1,w_2,\dots,w_n\in\Bbb{C}$ be arbitrary. Suppose $f, g$ are two polynomials of degree less than $n$ such that
$$f(z_j)=w_j,\qquad g(z_j)=\bar{w}_j \...

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**4**answers

1k views

### What can be said about this double sum?

Question. Can this number be expressed in terms of classical values?
$$\sum_{n,m=1}^{\infty}\frac1{(n^2+m^2)^{\frac32}}=1.056348517615643291\dots$$
UPDATE. I'm encouraged by Noam, Kevin and Igor's ...

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votes

**2**answers

306 views

### what is this sum of squares of algebraic functions?

This question is inspired by the MO query here, although it has no direct implications.
Define the family of polynomial functions
$$f_n(x)=n^2x^{n-1}-\frac{d}{dx}\left(\frac{x^n-1}{x-1}\right),$$
and ...

**4**

votes

**3**answers

682 views

### roots of higher derivatives of exponential

Consider the Gaussian function $f(z)=e^{-z^2}$ which has no zeros on the complex domain. Let $D$ denote derivative w.r.t. the variable $z$.
Question. Is it true that $D^nf(z)=0$ has only real roots ...

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**3**answers

4k views

### Ideas in the elementary proof of the prime number theorem (Selberg / Erdős)

I'm reading the elementary proof of prime number theorem (Selberg / Erdős, around 1949).
One key step is to prove that, with $\vartheta(x) = \sum_{p\leq x} \log p$,
$$(1) \qquad\qquad \vartheta(x) \...

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votes

**1**answer

500 views

### Beauty of some numbers discovered by Ramanujan

I am a graduate PhD student and my topic is analytic number theory. I am also a mathematics teacher. I am planning to give a course to pupils in high school that motivates them to study arithmetic and ...

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votes

**1**answer

91 views

### Under what condition does Courant–Fischer–Weyl min-max principle hold in general?

From Wikipedia:
Let $A$ be a $n \times n$ Hermitian matrix. As with many other variational results on eigenvalues, one considers the Rayleigh–Ritz quotient $RA : C^n \backslash \{0\} \to \mathbb{R}$ ...

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**1**answer

159 views

### Homogeneous subsets of the sphere

Let $S$ be a (unit) sphere in a Hilbert Space $H$ with $\dim H \ge 3$. Let $A \subset S$ have the following properties:
$A$ is connected;
The affine hull of $A$ is the whole space;
For every $x,y\in ...

**1**

vote

**0**answers

104 views

### Carry operations when adding two numbers [closed]

Let $x$ be a large positive real number and let $q\leq 2$ be a positive integer. It is known that for a positive integer $n,$ there exists a unique sequence $\left\{0\leq n_k\leq q-1\right\}_{k\geq 0}$...

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**0**answers

93 views

### Digits of sums of two integers [closed]

Let $q$ be a non-negative integer $\geq 2.$ For a non-negative integer $n$ It is known that there exixts a unique sequence of integer $0\leq n_k \leq q-1$ such that $$n=\sum_{k=0}^{+\infty} n_k q^k.$$
...

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**4**answers

319 views

### Binomial ID $\sum_{k=m}^p(-1)^{k+m}\binom{k}{m}\binom{n+p+1}{n+k+1}=\binom{n+p-m}{n}$

Could I get some help with proving this identity?
$$\sum_{k=m}^p(-1)^{k+m}\binom{k}{m}\binom{n+p+1}{n+k+1}=\binom{n+p-m}{n}.$$
It has been checked in Matlab for various small $n,m$ and $p$. I have a ...

**7**

votes

**1**answer

199 views

### Five cubes, Hadamard and Shklyarskiy

Here is my(=bad) translation of from the paper about Shklyarskiy by Golovina:
... in 1937/38 Dodik presented to school students a complete proof of Abel's theorem about equations of degree 5. He ...

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**0**answers

142 views

### Minkowski's lattice theorem in fragments of arithmetic

It is widely remarked that Minkowski's lattice theorem (or, convex body theorem) is a kind of geometrical pigeonhole principle. And it seems it should have a very elementary proof at least for convex ...

**3**

votes

**1**answer

186 views

### Independence of radicals: First-principles proof of special case

Reposting from MathStackexchange, original post is here, but got no answer.
I've known this problem for a long time:
Problem. Show that the number $\alpha=\sqrt{1} + \sqrt{2} + \ldots + \sqrt{n}$ is ...