2
votes
1answer
33 views

is my induction proof sufficient?

question; prove that $\forall\ n\ge4, n\in \mathbb{Z}, \ n!\gt n^2$. my work; let $n=4$ then $4!=24 \gt 4^2=16.$ true. now assume $n! \gt n^2$ is true for all $n\le k$ so now assume $k! \gt ...
0
votes
1answer
40 views

An Inequality Involving Prime Numbers

Let $p_i$ be the $i^{th}$ prime number. It seems as though the following inequality is true for all positive integers $m$ and real numbers $x>1$: ...
1
vote
0answers
19 views

An inequality involving Möbius function [duplicate]

For any positive integer $n$ show the inequality holds : $$\left|\sum_{i=1}^{n}\frac{\mu(i)}{i}\right|\le 1$$ I tried induction. when $\mu(n+1)=0$ it is trivial. But what if $\mu(n+1)\ne 0$? I am ...
1
vote
2answers
61 views

How to establish this inequality: $(1-a)(1-b)(1-c) \geq 8abc$ for $a+b+c=1$?

Let $a$, $b$, and $c$ be positive real numbers such that $a+b+c = 1$. Then how to establish the following inequality? $$ (1-a)(1-b)(1-c) \geq 8abc.$$ My effort: Since $a+b+c =1$, we can write $$ ...
2
votes
1answer
44 views

What is the most elementary proof of these inequalities?

Let $p$ be a non-zero integer, and let $x_1$, $\ldots$, $x_n$ be $n$ positive real numbers. Then we define the $p$-th power mean $M_p$ of these numbers as $$ M_p \colon= (\frac{x_1^p + \ldots + ...
1
vote
2answers
69 views

How to establish this inequality without using induction?

Given the Fibonacci sequence $a_1 = 1$, $a_2 = 2$, $\ldots$, $a_{n+1} = a_n + a_{n-1} $ for $n \geq 2$, how to derive, without using induction, the inequality $$ a_n < (\frac{1+\sqrt{5}}{2})^n $$ ...
1
vote
0answers
29 views

Can we prove this inequality in another way?

As explained here, I've managed to prove the following inequality: $\sigma(n)\geq\sqrt n(d(n)-2)+n+1$. This can be proved easily in two cases (one for $n$ being a perfect square and one for otherwise) ...
2
votes
0answers
41 views

Proving a quantity negative.

For $j\in{1,2,3}$ let $x_j,y_j \in R$ be nonzero and let $v_j=x_j+y_j$. Suppose that following holds: $$x_1x_2x_3=−y_1y_2y_3 \quad \text{and} \quad x^2_1+x^2_2+x^2_3=y^2_1+y^2_2+y^2_3$$ nd that ...
1
vote
1answer
26 views

which continued fraction is bigger? $[1,1,a,1,1,1,1]$ or $[1,1,1,b,1,1,1]$

Let's say I have a continued fraction $a = [a_1, a_2, \dots, a_n]$ but I make a mistake and switch the digit at two places, do I get a number which is bigger or smaller? For $a,b \in \mathbb{N}$ ...
3
votes
1answer
94 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 ...
13
votes
2answers
362 views

Prove that: $x_1\cdot x_2\cdots x_n>y_1\cdot y_2\cdots y_m$.

For two positive integer sequences $x_1,x_2,\ldots,x_n$ and $y_1,y_2,\ldots,y_m$ satisfying $x_i\neq x_j\quad \text{and}\quad y_i\neq y_j\quad \forall i,j, i \ne j$ ...
0
votes
1answer
35 views

An unusual inequality

Problem: $(x_i)_{i=1}^n$ is a finite sequence of positive integers. Define $f\big(S\big)=\displaystyle \sum_{i\,\in\, S\,\subseteq\, [n]}x_i$, and suppose $f$ is injective. Prove that: ...
1
vote
1answer
104 views

Proof of $x^y<y^x$ when $e\le y<x$ without calculus

I know that you can prove $e\le y<x$ implies $x^y<y^x$ by the following method: $$x^y<y^x\iff y \log x<x\log y\iff \frac{\log x}x<\frac{\log y}y,$$ and since $\frac d{dx}\frac{\log ...
0
votes
1answer
52 views

Bounds on functions using inequalities?

I'm studying inequalities as part of a course on Numbers, Proofs and Mathematical Induction. There is one type of question that I don't understand, primarily because there's only one example in the ...
4
votes
1answer
53 views

If distinct numbers $a,b,c\in\mathbb N^+$ satisfy $(a+b)(a+c)=(b+c)^2$, prove that $(b-c)^2>8(b+c)$.

If distinct numbers $a,b,c\in\mathbb N^+$ satisfy $$(a+b)(a+c)=(b+c)^2$$prove that $$(b-c)^2>8(b+c).$$ The first thing I did after I saw the problem was turning the inequality into this: ...
3
votes
2answers
61 views

Let $a_1,a_2,\ldots,a_{100}$ be real numbers,each less than one,satisfy $a_1+a_2+\ldots+a_{100} >1$.

Let $a_1,a_2,\ldots,a_{100}$ be real numbers,each less than one,satisfy $$a_1+a_2+\ldots+a_{100}>1$$ Prove the following statements: (i) Let $n_0$ be the smallest integer $n$ such that ...
2
votes
1answer
38 views

Is $\frac{k!!}{j!!(k-j)!!}\leq\frac{k!}{j!(k-j)!}$ for all integers $j$ and $k$, where $0\leq j\leq k$?

For all integers $j$ and $k$, where $0\leq j\leq k$, is the inequality $\frac{k!!}{j!!(k-j)!!}\leq\frac{k!}{j!(k-j)!}$ true? I have a feeling that it is and it would be helpful to me if it is, ...
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 ...
1
vote
1answer
53 views

Let $a_1,a_2,\ldots,a_{100}$ be real numbers,each less than one,satisfy $a_1+a_2+\ldots+a_{100} >1$.Prove the following statements

Let $a_1,a_2,\ldots,a_{100}$ be real numbers,each less than one,satisfy $$a_1+a_2+\ldots+a_{100}>1$$ Prove the following statements: (i) Let $n_0$ be the smallest integer $n$ such that ...
4
votes
2answers
100 views

If $a_1+a_2+\ldots+a_{2000}>a_1a_2\ldots a_{2000}$, prove that at least $1990$ of those numbers are equal to $1$.

If $a_1,a_2,\ldots,a_{2000}\in\mathbb N$ and$$a_1+a_2+\ldots+a_{2000}>a_1a_2\ldots a_{2000}$$ Prove that at least $1990$ of those numbers are equal to $1$. That's an unusual problem for me and I ...
1
vote
1answer
84 views

If $m,n\in \mathbb N$ and $n>m$, prove that $lcm(m,n)+lcm(m+1,n+1)>\frac{2mn}{\sqrt{n-m}}$.

Where $lcm$ is the least common multiple. I've changed it to: $$\frac{mn}{\gcd(m,n)}+\frac{(m+1)(n+1)}{\gcd(m+1,n+1)}>\frac{2mn}{\sqrt{n-m}}$$ Can't see how to continue. Is there a way to ...
1
vote
4answers
149 views

Which is larger :: $y!$ or $x^y$, for numbers $x,y$.

This is a generalization of this question :: Which is larger? $20!$ or $2^{40}$?. No explicit general solution was presented there and I'm just curious :D Thank-you. Edit :: I want a most-general ...
0
votes
1answer
29 views

Prove that there exist two infinite sequences that simultaneously satisfies all these conditions

Prove that there exist two infinite sequences $\langle a_n\rangle_{n\geq 1}$ and $\langle b_n\rangle_{n\geq 1}$ of positive integers such that the following conditions hold simultaneously: $$1 < ...
5
votes
1answer
146 views

How to prove this inequality $\pi(x) > \log x - 1$ involving the prime counting function?

Problem Prove that $\pi(x) > \log x - 1$. Progress Based on a hint and very elementary methods, I got that $$ \prod_{p \leq x} (1-p^{-1})^{-1} \leq \prod_{k=2}^{\pi(x)+1} (1-k^{-1})^{-1}. $$ The ...
5
votes
2answers
286 views

Proving Inequality with the Greatest Integer Function

Show that $$[(m+n)x]+[(m+n)y] \ge [mx+(n-1)y]+[my+(n-1)x]$$ where $m,~n \in \Bbb{N}$ and $0\le x,~y < 1$. I've tried everything for about half a day and still couldn't figure it out. ...
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 ...
4
votes
1answer
93 views

Does every record of the arithmetic derivative of natural numbers occur at a practical number?

Consider the arithmetic derivative of natural numbers, as defined here. By this definition, for every integer $n>1$, with canonical prime factorization ...
6
votes
2answers
137 views

Simple Divisor Summation Inequality (with Moebius function)

Show that $$\left| \sum_{k=1}^{n} \frac {\mu(k)}{k} \right| \le 1 $$ where $\mu$ is Moebius function and n is a positive integer. The hard thing here is that the sum is not directly ...
1
vote
1answer
75 views

Determine whether $\sigma(n)<e^\gamma n \omega(n)$ for all $n$ not of the form $2^x$

Determine whether $\sigma(n)<e^\gamma n \omega(n)$ for all $n$ not of the form $2^x$. In words (to define the symbols), the sum of the divisors of $n$ is less than the product of Euler's number to ...
1
vote
0answers
39 views

Ineqality regarding LCM of $1, 2, \ldots, n$

While going through F. Beukers proof of irrationality of $\zeta(3)$ I found the inequality $d_{n} < 3^{n}$ for all sufficiently large values of $n$ where $d_{n}$ denotes the LCM of all the numbers ...
2
votes
2answers
265 views

Proving that if $2a + 3b \ge 12m + 1$, then $a \ge 3m + 1$ or $b \ge 2m + 1$ [duplicate]

Let $a$, $b$, $m$ be integers. Prove that if $2a + 3b \ge 12m + 1$, then $a \ge 3m + 1$ or $b \ge 2m + 1$. I need help proving this. I am not sure what to do. Thank you for all of the edits. ...
2
votes
2answers
53 views

Prove modular inequalities $ab + ac\le a(b+ac)$ and $(a+b)(a+c)\ge a+b(a+c)$

How to prove $$(a\cdot b)+(a\cdot c)\le a\cdot\big(b+(a\cdot c)\big)$$ and $$(a+b)\cdot(a+c)\ge a+\big(b\cdot(a+c)\big)\;?$$ I have tried this. Using distributive property, I think we can get ...
2
votes
3answers
82 views

How to prove $(1+x)^n\geq 1+nx+\frac{n(n-1)}{2}x^2$ for all $x\geq 0$ and $n\geq 1$?

I've got most of the inductive work done but I'm stuck near the very end. I'm not so great with using induction when inequalities are involved, so I have no idea how to get what I need... ...
0
votes
0answers
45 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 + ...
0
votes
0answers
108 views

Can an odd perfect number be divisible by 101?

Preamble - This question is an offshoot from the following earlier questions here at MSE: Can an odd perfect number be divisible by 825? Can an odd perfect number be divisible by 165? Odd perfect ...
1
vote
2answers
162 views

Does the following inequality hold if and only if $N$ is an odd deficient number?

Let $N \in \mathbb{N}$. (That is, let $N$ be a positive integer.) This is in reference to two of my earlier questions here at MSE: Does the following inequality hold true, in general? Does this ...
4
votes
0answers
177 views

A question about odd perfect numbers

Edit [in response to a comment from anon]: Hereinafter, $N$ is a positive integer, $\sigma(N)$ is the sum-of-divisors of $N$, $\omega(N)$ is the number of distinct prime factors of $N$, and ...
1
vote
2answers
151 views

Does this inequality hold true, in general?

Let $$N = \prod_{i=1}^{\omega(N)}{{p_i}^{\alpha_i}}$$ be the prime factorization of the positive integer $N$. Does the following inequality hold true in general? ...
2
votes
1answer
148 views

Floor Inequalities

Proving the integrality of an fractions of factorials can be done through De Polignac formula for the exponent of factorials, reducing the question to an floored inequality. Some of those inequalities ...
6
votes
4answers
283 views

Greatest integer $n$ where $n \lt (\sqrt5 +\sqrt7)^6$

I'm really not sure how to do this. I factored out a power of $3$ and squared so that I have $2^3 (6+\sqrt{35})^3 \gt n$ , and I know that if I can prove that $12^3-1 \le (6+\sqrt{35})^3 \lt 12^3$ I ...
1
vote
1answer
90 views

Proving the existence of integer solutions to linear inequalities

Let $b_k\in\mathbb{Z}^n$ for $1\le k\le m$ for some $m,n$. I wish to prove the existence of two vectors $x,y\in\mathbb{Z}^n$ such that for all $k$, $b_k\cdot x\ne 0$ and $b_k\cdot y\ne 0$ with ...
4
votes
1answer
110 views

Finding an upper bound on a fraction

$0<\varepsilon <1$. If $n_k$ and $a_k$ are positive integers for which $$n_{k+1}=a_{k+1}n_k+n_{k-1}$$ Let $L\in\mathbb{N}.$ If $L>a_k \ge 3$, what's the smallest upper bound I can place on ...
2
votes
1answer
63 views

Does this inequality have any solutions for composite $n \in \mathbb{N}$?

Does this inequality have any solutions for composite $n \in \mathbb{N}$? $$\sqrt{2} < \frac{\sigma_1(n^2)}{n^2} < \frac{4n^2}{(n + 1)^2}$$ Note that $\sigma_1$ is the sum-of-divisors ...
3
votes
1answer
99 views

Does this inequality have any solutions in $\mathbb{N}$?

Does this (number-theoretic) inequality have any solutions $x \in \mathbb{N}$? $$\frac{\sigma_1(x)}{x} < \frac{2x}{x + 1}$$ Notice that we necessarily have $x > 1$.
1
vote
2answers
81 views

Find the greatest integer $N$ such that…

Find the greatest integer $N$ such that $N<\dfrac{1}{\sqrt{33+\sqrt{128}}+\sqrt{2}-8}$. The way I did it is this: first, I rewrote the biggest square root as $\sqrt{1+2*16+8\sqrt{2}}$. Then I ...
2
votes
1answer
59 views

Prove that $n^2 \leq \frac{c^n+c^{-n}-2}{c+c^{-1}-2}$.

Let $c \not= 1$ be a real positive number, and let $n$ be a positive integer. Prove that $$n^2 \leq \frac{c^n+c^{-n}-2}{c+c^{-1}-2}.$$ My initial thought was to try and induct on $n$, but the ...
3
votes
1answer
152 views

Average of divisors of n.

Let n be a natural number and let $f(n)=\frac{\sigma(n)}{d(n)}$ be the arithmetical average of n's divisors. Either prove or give a counterexample that for all natural numbers like n, which are not ...