# Tagged Questions

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### 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 ...
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### 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. ...
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### 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 $d(n)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $$\sqrt{n} +\sqrt{n+1} <\sqrt{x} +\sqrt{y} <\sqrt{4n+2}.$$
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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
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### 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 ...
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### How to prove that the partial Euler product of primes less than or equal x is bounded from below by log(x)? [closed]

How does one prove $\prod_{p \leq x}(1 - \frac{1}{p})^{-1} \geq \log(x)$?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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: $$\Longleftrightarrow(n\sqrt{2009}-m)^2k^2n^2>1$$ ...
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### 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 ...
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### 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 ...
### 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 ...
### 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 + ...