1
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1answer
46 views

Regarding 'non-square- free ' numbers.

Call an integer 'n' that is not a square or a prime power or a square-free a 'square-in'.Let n be square-in. Then between n and (2 n) is there another square-in? This is a kind of 'variation' on ...
1
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2answers
29 views

Regarding square-free numbers and their doubles.

Is it true that between any non-prime square-free number and it's double is another non-prime square-free number?
3
votes
3answers
76 views

Proof by induction: $n$th Fibonacci number is at most $ 2^n$

I'm trying to find the proof by induction of the following claim: For all $n\in\mathbb N$, $\operatorname{fibonacci}(n) \le 2^n$ My Proof: Base case: $n = 1$ $\operatorname{fibonacci}(1) \le 2^ 1$ ...
1
vote
3answers
26 views

Inequality involving floor function and fractions

I have little to no experience working with floor inequalities so I am kind of stuck on this one. It seems pretty intuitive though. So basically I want to show that ...
0
votes
0answers
38 views

A question related to the Descartes-Frenicle-Sorli conjecture on odd perfect numbers

Good day to everyone! I apologize in advance for the somewhat long post, but I had to put in all the details into a single question to communicate what I believe to be a viable approach to odd ...
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
41 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
20 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
66 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
30 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
42 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
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 ...
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
55 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
54 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
101 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
85 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
153 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
147 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
294 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
97 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
140 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
77 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
268 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. ...
3
votes
1answer
277 views

Use induction to prove that $ 1 + \frac {1}{\sqrt{2}} + \frac {1}{\sqrt{3}} … + \frac {1}{\sqrt{n}} < 2\sqrt{n}$

Use induction to prove that $ 1 + \frac {1}{\sqrt{2}} + \frac {1}{\sqrt{3}} ... + \frac {1}{\sqrt{n}} < 2\sqrt{n} $ My attempt was as follows: Lets assume the inequality is true for n = k $S_k ...
2
votes
2answers
54 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
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 + ...
0
votes
0answers
127 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
164 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
180 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
154 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
150 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
92 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 ...