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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 ...

9
votes
2answers
494 views

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
votes
2answers
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
votes
0answers
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
0answers
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
votes
0answers
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
votes
1answer
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
1answer
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
votes
1answer
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
1answer
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
votes
1answer
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
votes
2answers
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 $...
6
votes
2answers
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
votes
0answers
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
votes
1answer
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
vote
0answers
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
votes
2answers
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
votes
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
2answers
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 \...
1
vote
0answers
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
0answers
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
1answer
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
1answer
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
2answers
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
1answer
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
1answer
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
1answer
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
1answer
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
votes
0answers
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
3answers
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
votes
2answers
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
votes
1answer
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
2answers
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
votes
4answers
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
2answers
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
3answers
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
3answers
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
1answer
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
1answer
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
1answer
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 ...