Number Theory : Find the group of A such that $A=\{x \in \mathbb{Z}:\frac{x^3-3x+2}{2x+1}\in \mathbb{Z}\}$

Question

Find the group A such that $A=\{x \in \mathbb{Z}:\frac{x^3-3x+2}{2x+1}\in \mathbb{Z}\}$ ?

Polynomial Long Division we get $\frac{x^{2}}{2}-\frac{x}{4}-\frac{11}{8}+\frac{27}{8\left(2x+1\right)}$

but how i can from here find all x such that $x \in \mathbb{Z}$ ??

i did use the hint for $y = 2x+1$ so that $x =\frac{y-1}{2}$

made the equation to be $\frac{(y-1)^3}{8}-\frac{3(y-1)}{2}+2=\frac{y^3-3y^2-9y+27}{8}$

$\frac{\frac{y^3-3y^2-9y+27}{8}}{y}=\frac{y^3-3y^2-9y+27}{8y}$

I can not understand with the help of the clues, can some 1 some formal proof ?

Well, a natural first step might be to try some examples. Have you found some integers in $A$? — lulu, Nov 06 '20 at 17:24
What happens to the last fraction and the sum if $(2x+1)$ is not a divisor of $27$? — player3236, Nov 06 '20 at 17:25
@lulu of course i found some integers in A but i want to see way . — ATB, Nov 06 '20 at 17:50

score 1 · Answer 1 · answered Nov 06 '20 at 17:25

1

Hint: Since $2x+1$ is odd for all integers $x$, we have that $\frac{x^3-3x+2}{2x+1}\in\mathbb Z$ if and only if $\frac{2(x^3-3x+2)}{2x+1}\in\mathbb Z$.

answered Nov 06 '20 at 17:25

Ennar

24,364

from where i can know that if and only if $\frac{2(x^3-3x+2)}{2x+1}\in \mathbb{Z}$ ?
so the next equation is $\frac{x^3-3x+2}{x+\frac{1}{2}}\in \mathbb{Z}$ ??

ATB

Nov 06 '20 at 18:05

The New Polynomial is $x^2-\frac{x}{2}-\frac{11}{4}+\frac{27}{4\left(2x+1\right)}$ just from where to search what number can be divided to $x \in \mathbb{Z}$ ?? – ATB Nov 06 '20 at 18:14

One direction is obvious, if you multiply an integer by $2$, you get an integer. The other follows from the fact that $2x+1$ and $2$ are relatively prime, so if $2x+1$ divides $2(x^3-3x+2)$, it must divide $x^3 - 3x + 2$.

– Ennar Nov 06 '20 at 18:19

lab bhattacharjee · Answer 2 · 2020-11-06T18:40:00.777

1

Hint:

If $2x+1=y$

$$x^3-3x+2=(y-1)^3/8-3(y-1)/2+2=\dfrac{y^3-3y^2-9y+27}8$$ which will be divisible by $y$

$\iff y$ divides $y^3-3y^2-9y+27$ as $y$ is odd

edited Nov 06 '20 at 18:40

answered Nov 06 '20 at 18:21

lab bhattacharjee

279,016

Didn't check your work, but according to mine, seems to me that it should be $27$, not $29$, right? – Ennar Nov 06 '20 at 18:30
@Ennar, thanks for your observation. Actually after submission, the latex was not rendered properly. – lab bhattacharjee Nov 06 '20 at 18:41
@labbhattacharjee thanks , i use your hint when i divides by $y$ its by $8y$ and its even right ? so how can make step forward i dont get it – ATB Nov 06 '20 at 18:53
@atb, It's clear that $y$ has to divide $27$ .Now we need to check for all the possible values of $y$ – lab bhattacharjee Nov 06 '20 at 19:06
i just need to find all the number divided by 27 because i got 8y and i need to find all the numbers such that 27 divided by 8? maybe can write small explanation :D? – ATB Nov 06 '20 at 20:17
@atb. If $x\ge1, y\ge3$ so, the options for $y$ are $$3,9,27$$ example, if $y=27, x=13$ $$x^3-3x+2=13(13^2-3)+2, 2x+1=?$$ – lab bhattacharjee Nov 08 '20 at 15:04

DanielWainfleet · Accepted Answer · 2020-11-07T10:35:13.877

0

$2x+1|x^3-3x+2\iff 2x+1|8(x^3-3x+2)$

$\iff 2x+1|8(x^3-3x+2)-(2x+1)^3=-12x^2-30x+15$

$\iff 2x+1|3(2x+1)^2-12x^2-30x+15=-18x+18$

$\iff 2x+1|9(2x+1)-18x+18=27$

$\iff x\in \{-14,-5,-2,-1,0,1,4,13\}.$

edited Nov 07 '20 at 10:35

answered Nov 07 '20 at 10:30

DanielWainfleet

59,529

Thanks , this is the full sol for the problem ? @DanielWainfleet – ATB Nov 07 '20 at 13:30
1

Yes. But don't call it a group, as "group" has a special meaning in algebra. – DanielWainfleet Nov 07 '20 at 18:17
what its mean the right and left arrow ? if and only if ? you can maybe write it like a formal proof ? – ATB Nov 07 '20 at 19:26
thank you for the proof , but i dont understand why u use the right and left arrow you can edit it for made its formal proof please? – ATB Nov 07 '20 at 20:17
$\iff$ means "if and only if". In LaTex it is \iff. And "iff" is often used in plain text for "if and only if". What I wrote $is$ formal. – DanielWainfleet Nov 08 '20 at 00:02
simple question why if u multiply the equation by 8 its must divided the 2x+1 ? – ATB Nov 08 '20 at 10:49
$2x+1$ is odd so $2x+1$ divides $y$ iff $2x+1$ divides $2y, 4y, 8y,$ etc. – DanielWainfleet Nov 08 '20 at 16:30
thank u. Any chance on please elaberationg on ur answer , because i didn't understand the changes between every line and the steps of the equation. – ATB Nov 08 '20 at 17:15
your proof its very good but if u can explain each step because i didnt understand the changes please . – ATB Nov 09 '20 at 11:53
If B\ne 0 then B|C means C/B\in \Bbb Z. So if B|D then (B|C\iff B|C\pm D). Consider B=2x+1 and D=E(2x+1)^n with E\in\Bbb Z and n\in \Bbb N – DanielWainfleet Nov 12 '20 at 04:38

Number Theory : Find the group of A such that $A=\{x \in \mathbb{Z}:\frac{x^3-3x+2}{2x+1}\in \mathbb{Z}\}$

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