# 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 ...

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### Certain matrices of interesting determinant

Let $M_n$ be the $n\times n$ matrix with entries
$$\binom{i}{2j}+\binom{j}{2i}, \qquad \text{for $1\leq i,j\leq n$}.$$
QUESTION. Is this true? There is some evidence. The determinant $\det(M_{2n+1})...

**17**

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**2**answers

2k 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 ...

**0**

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**0**answers

61 views

### Relation between primeness, coprimeness, totient, and gcd function

There are two number theoretic facts that seem to be unrelated at very first sight but at second sight seem to be strongly related to each other:
(1) Primeness of one number and coprimeness of two ...

**1**

vote

**0**answers

47 views

### Supremum of an almost surely continuous random process

I was learning this proposition
and now I have a question, how to prove it for an almost surely continuous process? I would be very grateful for any tips.

**0**

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**0**answers

41 views

### What can we say about the Bargmann transform of bounded function?

Let $f, g\in \mathcal{S}(\mathbb R)$. Then the convolution of two Schwartz class function is again Schwartz class function, that is, $f\ast g \in \mathcal{S}(\mathbb R).$
Now we define
$$ H(t)= H(...

**10**

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**1**answer

234 views

### A set of prime numbers

Consider a non-empty set $S$ of primes, with the property that, for every finite subset $S'\subset S$, all the primes dividing $\left(\prod_{k\in S'}k\right)+1$ are in $S$.
For instance, it can ...

**2**

votes

**1**answer

76 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 $k\times k$ ...

**3**

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**1**answer

269 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

203 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 ...

**5**

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**1**answer

119 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 ...

**7**

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**2**answers

179 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

132 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 $\lambda'=(3,2,2,1)$.
...

**13**

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**0**answers

216 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**

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**1**answer

940 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\...

**0**

votes

**0**answers

415 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}\...

**1**

vote

**1**answer

345 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 ...

**0**

votes

**1**answer

149 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 ...

**9**

votes

**1**answer

574 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)...

**3**

votes

**1**answer

276 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**

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**0**answers

133 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 ...

**7**

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**2**answers

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

**-1**

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**1**answer

133 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

227 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?

**3**

votes

**1**answer

230 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}\...

**8**

votes

**1**answer

413 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 $\frac{2k+1}{n+k+1}\binom{2n}{n-k}$ ...

**8**

votes

**1**answer

954 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

232 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 :
...

**1**

vote

**1**answer

190 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 ...

**2**

votes

**2**answers

410 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 \...

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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+...

**2**

votes

**0**answers

62 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

181 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(...

**2**

votes

**1**answer

213 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**

votes

**1**answer

250 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

289 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

50 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 $...

**15**

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**0**answers

577 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 ...

**5**

votes

**3**answers

939 views

### Solution to a Diophantine equation

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

**3**

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**2**answers

613 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 ...

**5**

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**1**answer

573 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 $...

**3**

votes

**2**answers

257 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 \...

**17**

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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 ...

**4**

votes

**2**answers

307 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

684 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 ...

**23**

votes

**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) \...

**5**

votes

**1**answer

508 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 ...

**0**

votes

**1**answer

104 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 $RA : C^n \backslash \{0\} \to \mathbb{R}$ ...

**2**

votes

**1**answer

160 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 ...