1
vote
0answers
53 views

Comparing a number with a line of power

How do you compare which is bigger (or maybe equal), LHS or RHS, in $$a \sim b_1^{b_2^{.^{.^{.^{b_n}}}}}$$ given $a$ and $b_i$, $1 \leq i \leq n$, are non-negative integers (also could be big)? The ...
2
votes
0answers
72 views

Diophantine inequality that comes up after Vieta Jumping Hurwitz technique

I am blaming this on Prove the equality EDITTTTT: allowing $x_1 \geq x_2$ and $x_2 \geq x_n,$ I would rather not explain what that was about and the only changes are in $n=3,4,$ already settled. ...
0
votes
0answers
13 views

Is there an effective bound known for the coefficients of half integer weight cusp forms?

If $f(z)=\sum a_n q^n$ is a cusp form (of integer weight) normalized so that $a_1=1$, we have the inequality $$\vert a_n \vert \leq d(n) n^{(k-1)/2},$$ known as the Deligne bound (in which $d(n)$ ...
3
votes
0answers
56 views

An Inequality with Prime Numbers

Let $m\geq 8$ be an integer, and let $q$ be the smallest prime that is greater than $m$. Let $p$ be a prime that satisfies $q\leq p<q^2$, and let $p_0$ be the smallest prime such that ...
16
votes
4answers
299 views

An inequality on sequences with each term dividing sum of two neighbouring terms

Positive integers $x_1, x_2, \dots, x_n$ ($n \ge 4$) are arranged in a circle such that each $x_i$ divides the sum of the neighbors; that is $$\frac{x_{i-1}+x_{i+1}}{x_i} = k_i $$ is an integer for ...
0
votes
0answers
124 views

Prove the inequality $\sum_{i,j = 1}^n {{A_{i,j}}({x_i}^2 - {x_i}{x_j})} \ge 0 $

A is a square matrix with positive elements and x is a real vector (both of them n>1 dimensional). Prove that for any such matrix and vector $$\sum\limits_{i,j = 1}^n {{A_{i,j}}({x_i}^2 - {x_i}{x_j})} ...
0
votes
0answers
26 views

An inequality with $\omega(n)$ [duplicate]

Prove: For any positive integer $k, N$, $$\left(\frac{1}{N}\sum\limits_{n=1}^{N}\left(\omega (n)\right)^k\right)^{\frac{1}{k}}\leq k+\sum\limits_{q\leq N}\frac{1}{q}$$ Where $\sum\limits_{q\leq ...
1
vote
0answers
136 views

An number theoretic inequality

Prove this inequality : $$\displaystyle \prod_{i\le \left\lfloor{\frac{n-1}{2}}\right\rfloor}\left\lfloor{\frac{\left\lfloor{\frac{n-1}{2}}\right\rfloor}{i}}\right\rfloor\le ...
2
votes
0answers
61 views

Prove the inequality for composite numbers

Is it true that $c_m+c_n$ $>$ $c_{m+n}$ for all $m$, $n$ $\in$ $\mathbb{N}$? Though the result seems true, I can't get a solution. Even the bounds on $c_n$ obtained from Prime Number Theorem ...
3
votes
1answer
98 views

Non-existence of natural numbers such that $\sqrt{n} +\sqrt{n+1} <\sqrt{x} +\sqrt{y} <\sqrt{4n+2}$

Show that for any $n\in\mathbb{N}$ there does not exist natural numbers $x,y$ such that $$\sqrt{n} +\sqrt{n+1} <\sqrt{x} +\sqrt{y} <\sqrt{4n+2}.$$
0
votes
1answer
37 views

Unindentified inequality from Hardy-Littlewood-Polya

I found this while trying to understand a theorem. Could anyone tell me which famous inequality is being mentioned here, and where I can find a proof/ statement of that inequality? The article refers ...
10
votes
1answer
304 views

A unsolved puzzle from Number Theory/ Functional inequalities

The function $g:[0,1]\to[0,1]$ is continuously differentiable and increasing. Also, $g(0)=0$ and $g(1)=1$. Continuity and differentiability of higher orders can be assumed if necessary. The ...
0
votes
0answers
44 views

How prove this inequality $\frac{2n^2+1}{3n}\le \sum_{k=1}^{n}\frac{a(k)}{k}\le\frac{2n(n+3)}{3(n+1)}$

Question: For a postive integer $k$, let $a(k)$ be the largest odd divisor of $k$, show that : for each positive integer $n$, $$\dfrac{2n^2+1}{3n}\le ...
5
votes
0answers
62 views

Finding All Integers Satisfying the Condition

Find all the solutions of the inequality- $$\sqrt{x(\ln x +\ln \ln x)}-1 > y > \sqrt{x(\ln x+ \ln \ln x-1)}$$ Where $x,y$ $\in$ $\mathbb N$. Determine the set of integral values of $(x,y)$. ...
2
votes
0answers
60 views

All those unit fractions add to 1?

Consider $$S(n)=\{x \mid x=(a_1 ,a_2,a_3 \cdots a_n) \text{ where } \sum_{r=1}^{n}\frac{1}{a_r} =1 \}$$ Now let $|S(n)|$ denote the cardinaly (order) of set $S(n)$. Thus: $S(1)= \{(1)\} \implies ...
7
votes
2answers
120 views

On the numbers divisible by all the Integers not exceeding their $r^{th}$ roots.

Consider the set of all numbers which are divisible by all natural numbers not exceeding their square root, and denote this set by $S_2=\{1,2,3,4,6,8,12,24\}$ (Here the subscript indicates that we're ...
4
votes
1answer
50 views

On the sum of relatively prime number $<N$

Let $A(N)$ be a function which is the sum of all numbers relatively prime and $<N$ and $B(N)$ the sum of remaining $N−\phi(N)$ numbers. Then I have the following questions- Q-1 For what values of ...
2
votes
1answer
80 views

A Shorter Proof of Rosser's Theorem Without Using The Prime Number Theorem

While researching on the elementary proof of Bertrand's Postulate I came to know about a theorem of Rosser's which states that $p_n$ $>$ $n$ $\text{ln}$ $n$. I have seen Rosser's original proof and ...
6
votes
2answers
149 views

How can we prove $1>\sum_{k=1}^{n}\frac{\varphi{(k)}}{k}\ln{\frac{2^k}{2^k-1}}>1-\frac{1}{2^n}$

Let $\varphi$ be Euler's totient function, where $\varphi{(1)}=1$. Prove that for any postive integer $n$ we have ...
1
vote
1answer
54 views

Why does this equation work?

let $ P(x) := \sum_{p \leq x} Log [p]$, then we have $P(2^{k+1}) = \sum_{i=0}^k ( P(2^{i+1}) - P(2^i)) < 2 \cdot Log[2] \cdot (1 + 2 + 4 +... + 2^k) \leq 4 \cdot Log[2] \cdot 2^k$. Why does ...
2
votes
1answer
24 views

Proof-Writing $\theta(n) \le \theta(2^{k+1}) < 4*log[2n]$

At the end of this message there are two steps that I do not understand. The proof wants to show in the end that : *$\theta(n) \le \theta(2^{k+1}) < 4*log[2n]$ by definition we have ...
3
votes
3answers
148 views

Find the maximum value of $abc$

$a,b,c$ are three positive real numbers such that $ab+bc+ca=12$. Then find the maximum value of $abc$
3
votes
1answer
59 views

How prove this there exist positive integer sets $A$ such $|A+A|>|A-A|$

let positive integer $n\ge 8$ is given, show that: there exist sets $A$ with the set of is positive integer.such $|A|=n$,and such $$|A+A|>|A-A|$$ where $$A-A=\{a-b|a\in A,b\in A\},A+A=\{a+b|a\in ...
2
votes
3answers
118 views

Maximize $\sqrt{2x + 13} + \sqrt[3]{3y+5} + \sqrt[4]{8z+12}$

Given three non-negative (as pointed out by Calvin Lin) real numbers $x+y+z = 3$, find the maximum value of $\sqrt{2x + 13} + \sqrt[3]{3y+5} + \sqrt[4]{8z+12}$. (Source : Singapore Math Olympiad ...
0
votes
2answers
33 views

sequence $a_n = \lceil \sqrt{2}n \rceil $

I was trying to prove $\lceil \sqrt{2}n \rceil + \lceil \sqrt{2}m \rceil \geq \lceil \sqrt{2}(n+m) \rceil$ where $m,n\in \mathbb{z}$ Direct proof I tried but could not figure out. I tried fixing m ...
2
votes
3answers
169 views

Finding the values of $A,B,C,D,E,F,G,H,J$

Given that the letters $A,B,C,D,E,F,G,H,J$ represents a distinct number from $1$ to $9$ each and $$\frac{A}{J}\left((B+C)^{D-E} - F^{GH}\right) = 10$$ $$C = B + 1$$ $$H = G + 3$$ find (edit: ...
4
votes
2answers
108 views

Finding the value of $(bc-ad)(ac-bd)(ab-cd)$

Let $a,b,c,d$ be $4$ distinct non-zero integers such that $a+b+c+d = 0$. It is know that the number $$M = (bc - ad)(ac - bd)(ab-cd)$$ lies strictly between $96100$ and $98000$. Determine the value ...
5
votes
1answer
143 views

How prove this number theory inequality $\left(\dfrac{1}{N}\sum_{n=1}^{N}(\omega{(n)})^k\right)^{\frac{1}{k}}\le k+\sum_{q\le N}\frac{1}{q}$

show that: for any positive numbers $k$ and $N$, have $$\left(\dfrac{1}{N}\sum_{n=1}^{N}(\omega{(n)})^k\right)^{\frac{1}{k}}\le k+\sum_{q\le N}\dfrac{1}{q}$$ where $\displaystyle\sum_{q\le N}$ is ...
18
votes
4answers
306 views

Find the smallest k such that $n^k > \sum_{i=0}^{n-1} i^k$

Let $n \in \mathbb{N}$. Is it possible to find the smallest $k \in \mathbb{N}$ such that $$n^k > \sum_{i=1}^{n-1} i^k \ ?$$ It's easy to prove that such $k$ exist because: $$n^k > 1^k + 2^k ...
0
votes
1answer
29 views

Proving that a certain sequence is bounded from above

Let $p_1,p_2,p_3,..$ be the sequence of primes in increasing order ($p_1=2,p_2=3,...$) .Let $x_n$ be given by: ...
0
votes
1answer
35 views

$\frac{2}{\frac{a}{x}+\frac{b}{y}}\leq ax+by$

$\frac{2}{\frac{a}{x}+\frac{b}{y}}\leq ax+by$, where a+b=1 and $a,b,x,y>0$ real numbers. Any hints? part (a) was showing $\frac{2}{\frac{1}{x}+\frac{1}{y}}\leq \sqrt{xy}\leq \frac{x+y}{2}$. To ...
2
votes
0answers
95 views

Sum of inverse of pairwise square roots [closed]

Prove that $$\sum_{i=1}^{50} \frac{1}{\sqrt{2i-1}+\sqrt{2i}} \gt \frac 92$$
0
votes
0answers
49 views

Given a set of inequalities, use what algorithm to minimize one variable?

I have the following set of inequalities: $$C_1 \le y-T_1x_1 \lt C_1+D_1$$ $$C_2 \le y-T_2x_2 \lt C_2+D_2$$ $$C_3 \le y-T_3x_3 \lt C_3+D_3$$ where $x_1, x_2, x_3$ and $y$ are non-negative integer ...
7
votes
1answer
127 views

Inequality with four positive integers looking for upper bound

Umm. This comes from Diophantine quartic equation in four variables and will finish the most important part if it can be done. Four positive integers $w,x,y,z.$ One equation and two inequalities $$ ...
12
votes
1answer
141 views

A claim that a is a square

We have integers $a,b,c,d$ such that $a<b\le c<d$ and $ad=bc$ and $\sqrt{d}-\sqrt{a}\le 1$.Show that $a$ is a perfect square.This question is pretty unbelievable for me.anyway I don't know if I ...
0
votes
0answers
26 views

Exponential Diophantine Inequality

how would one go about solving inequality of the form $|a2^n-b2^k|>1 $ for $a,b \in R$ and $n,k \in Z$. Assume that $|a|>|b|$. Any help will be appreciated. Thank you
1
vote
1answer
23 views

Interesting continued fraction problem $|r_i-u_0/u_1|\le\frac1{k_ik_{i+1}}$

Let $u_0/u_1$ be a rational number in lowest terms, and write $u_0/u_1=\langle a_0, a_1,...,a_n\rangle$ in standard continued fraction notation. Show that if $0\le i<n$, then ...
-1
votes
1answer
96 views
14
votes
2answers
203 views

Prove $\log_5{30}<\log_8{81}$

It's easy to prove this by calculator or computer, and I wonder can we prove that $$\log_5{30}<\log_8{81}\tag 1$$ by pencil and paper ? Thanks in advance ! Edit: $(1)$ can be written as ...
0
votes
1answer
36 views

Equality question

Hi I'm a bit confused with this? $\frac{1}{x} < 0 \iff x\frac{1}{x} < x\times 0 =0 \iff 1 < 0$ This was another question that I saw which was $\frac{1}{x} < 0$ but when I multiplied by ...
3
votes
1answer
101 views

Lower bound on a number theoretic function

Let $n$ be a positive odd integer, let $$n_j = \Bigl\{\frac{n}{2^{j+1}}\Bigr\}\,,$$ where $\{x\}$ denotes the fractional part of $x$, and finally let $k = \lceil \log_2 n\rceil$. Consider the ...
4
votes
1answer
113 views

Generalization of an inequality $0\lt e^6-{\pi}^4-{\pi}^5\lt 0.00002$

Question : Is the following true? For any $n\in\mathbb N$, there exists a triple $(k,l,m)\ (k,l,m\in\mathbb N)$ such that $$0\lt e^k-{\pi}^l-{\pi}^m\lt{10}^{-n}.$$ Motivation : A friend ...
0
votes
0answers
34 views

Lower-Upper bounds on the cardinality of a bounded set

Let $S$ be a finite set which is a subset of $\{(\alpha ,\beta ):\alpha , \beta \in \mathbb{Z}, \alpha\geq 0, \beta \geq 0\}$ and $ T(x,y)=\sum_{(\alpha ,\beta ) \in S} h_{\alpha, \beta} ...
1
vote
1answer
138 views

$\sum_{k_1+k_2+\cdots+k_N=n,\ k_i\ge0\in\mathbb Z}\frac1{\prod_{j=1}^{N}\{(N-1)k_j+1\}}\le 1$ is true for any $n,N\in\mathbb N$?

Is the following true for any $n,N\in\mathbb N$? $$\sum_{k_1+k_2+\cdots+k_N=n,\ k_i\ge0\in\mathbb Z}\frac1{\prod_{j=1}^{N}\{(N-1)k_j+1\}}\le 1$$ Motivation : I've known the $N=3$ case. ...
0
votes
1answer
197 views

Regarding Chebyshev's theta function

It is known that $x\sim \theta \left ( x \right )$, where $$\theta \left ( x \right )= \sum_{p\leqslant x}\log p.$$ For all values of x for which it has been calculated, $x> \theta \left ( x ...
3
votes
2answers
99 views

How prove this inequality $|n\sqrt{2009}-m|>\dfrac{1}{kn}$

if $k>\sqrt{2009}+\sqrt{2010}$,show that for any positive integer numbers $m,n$, have $$|n\sqrt{2009}-m|>\dfrac{1}{kn}$$ My try: $$\Longleftrightarrow(n\sqrt{2009}-m)^2k^2n^2>1$$ ...
6
votes
2answers
183 views

How find this$\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+…+\frac{1}{{{p}_{n}}}<10$

Let ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ be the prime numbers less than ${{2}^{100}}$. Prove that $$\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+...+\frac{1}{{{p}_{n}}}<10$$ This problem is from this ...
0
votes
1answer
59 views

Relation of $e$ to other numbers…

I found the following result, When i was working on my calculator . $$x^y < y^x \quad ,x < y \quad \text{ for } x,y<e$$ $$x^y > y^x \quad ,x < y \quad \text{ for } x,y>e$$ I can't ...
0
votes
0answers
46 views

What proportion of the positive integers satisfy $I(n^2) < (1 + \frac{1}{n})I(n)$, where $I(x)$ is the abundancy index of $x$?

Let $\sigma(x)$ denote the classical sum-of-divisors function, and let $$I(x) = \frac{\sigma(x)}{x}$$ be the abundancy index of the positive integer $x$. My question is this: What proportion of ...
0
votes
1answer
86 views

What proportion of the positive integers satisfy $I(n) < \frac{2n}{n + 1} \leq I(n^2)$ < 2?

Let $$I(x) = \frac{\sigma(x)}{x}$$ be the abundancy index of the positive integer $x$. Note that $\sigma(x)$ is the classical sum-of-divisors function. For example, $$\sigma(12) = 1 + 2 + 3 + 4 + ...