Primes and certain unit fractions

Are there primes $p,q$ and a natural number $a$ such that $\frac{1}{p}+\frac{1}{q}=\frac{1}{a}$?

Are there primes $p,q$ and a natural number $a$ such that $\frac{1}{p}+\frac{1}{q}=\frac{1}{a}$?

Banguizb
2022-07-07
Answered

Primes and certain unit fractions

Are there primes $p,q$ and a natural number $a$ such that $\frac{1}{p}+\frac{1}{q}=\frac{1}{a}$?

Are there primes $p,q$ and a natural number $a$ such that $\frac{1}{p}+\frac{1}{q}=\frac{1}{a}$?

You can still ask an expert for help

asked 2022-06-25

Prove that $\frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d}$

I'm reading a introductory book on mathematical proofs and I am stuck on a question.

Let $a,b,c,d$ be positive real numbers, prove that if $\frac{a}{b}<\frac{c}{d}$, then $\frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d}$

I'm reading a introductory book on mathematical proofs and I am stuck on a question.

Let $a,b,c,d$ be positive real numbers, prove that if $\frac{a}{b}<\frac{c}{d}$, then $\frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d}$

asked 2022-04-23

Is it possible to have a fraction wherein the numerator and denominator are also fractions?

For example:

$\frac{\frac{3}{5}}{\frac{7}{8}}$

I was wondering if such a situation had a name, wherein both the numerator and the denominator of a fraction consist of fractions themselves. I was also wondering if this was something common, or at least in some way useful.

For example:

I was wondering if such a situation had a name, wherein both the numerator and the denominator of a fraction consist of fractions themselves. I was also wondering if this was something common, or at least in some way useful.

asked 2022-06-21

How can I shorten this expression? $\frac{-x\cdot {a}_{1}-x\cdot {a}_{2}-x\cdot {a}_{3}-u}{{a}_{4}}\cdot l$

You see, there are many $-x$ terms. How can I make this shorter? Can I write:

$x\phantom{\rule{thinmathspace}{0ex}}\frac{-{a}_{1}-{a}_{2}-{a}_{3}}{{a}_{4}}\cdot l-u\phantom{\rule{thinmathspace}{0ex}}\frac{l}{{a}_{4}}$

Is this right?

Regards.

You see, there are many $-x$ terms. How can I make this shorter? Can I write:

$x\phantom{\rule{thinmathspace}{0ex}}\frac{-{a}_{1}-{a}_{2}-{a}_{3}}{{a}_{4}}\cdot l-u\phantom{\rule{thinmathspace}{0ex}}\frac{l}{{a}_{4}}$

Is this right?

Regards.

asked 2021-12-27

How do you write $\frac{2}{5}$ as decimal?

asked 2022-09-21

$\frac{{a}^{3}}{{b}^{2}}+\frac{{b}^{3}}{{c}^{2}}+\frac{{c}^{3}}{{a}^{2}}\ge 3\phantom{\rule{thinmathspace}{0ex}}\frac{{a}^{2}+{b}^{2}+{c}^{2}}{a+b+c}$

Proposition

For any positive numbers a, b, and c,

I am requesting an elementary, algebraic explanation to this inequality. (I suppose the condition for equality is that a=b=c.) I am not familiar with symmetric inequalities in three variables. I would appreciate any references.

Proposition

For any positive numbers a, b, and c,

I am requesting an elementary, algebraic explanation to this inequality. (I suppose the condition for equality is that a=b=c.) I am not familiar with symmetric inequalities in three variables. I would appreciate any references.

asked 2022-05-22

Inequality - GM, AM, HM and SM means

I've got stuck at this problem :

Prove that for any $a>0$ and any $b>0$ the following inequality is true:

$3(\frac{{a}^{3}}{{b}^{3}}+\frac{{b}^{3}}{{a}^{3}})\ge \frac{a}{b}+\frac{b}{a}+4$

The first thing that I've thought was the AM-GM inequality (the extended version - heard that is also known as The power mean inequality):

$HM\le GM\le AM\le SM$

where $HM$, $GM$, $AM$, and $SM$ refer to the harmonic, geometric, arithmetic, and square mean, respectively. CBS(Cauchy - Buniakowsky - Schwartz) also come to my mind, but I think it isn't helpful in this case.

I would be greatful for some hints.

Thanks!

I've got stuck at this problem :

Prove that for any $a>0$ and any $b>0$ the following inequality is true:

$3(\frac{{a}^{3}}{{b}^{3}}+\frac{{b}^{3}}{{a}^{3}})\ge \frac{a}{b}+\frac{b}{a}+4$

The first thing that I've thought was the AM-GM inequality (the extended version - heard that is also known as The power mean inequality):

$HM\le GM\le AM\le SM$

where $HM$, $GM$, $AM$, and $SM$ refer to the harmonic, geometric, arithmetic, and square mean, respectively. CBS(Cauchy - Buniakowsky - Schwartz) also come to my mind, but I think it isn't helpful in this case.

I would be greatful for some hints.

Thanks!

asked 2022-03-30