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MCIII Number Theory 3.12 and 3.13

 
 
JinTina的头像
MCIII Number Theory 3.12 and 3.13
JinTina - 2023年12月29日 Friday 13:34
 

Hello,

Hello! I did 3.12 problem using two ways, both are very tedious:

1. Using the quadratic formula to get x in terms of floor(x), then using floor(x)≤x<floor(x)+1

2. Finding floor(x) in terms of x and using x-1<floor(x)≤x

Both of them involve radicals and one of them requires plugging in multiple numbers back into the equation.

Is there a less tedious way to do this?

For 3.13, I added floor(x)+floor(1/x) to both sides to get:

floor(x)+floor(1/x)+1=x+1/x

then by using the fact that [a]+[b]≤[a+b]                                 (floor(x)=[x])

(floor(x)+floor(1/x))+1(=x+1/x)≤floor(x+1/x)+1

so x+1/x≤floor(x+1/x)+1 which is trivial since (let y=x+1/x) y-1<floor(y)≤y, add one to both sides y<floor(y)+1

When I get an inequality that is trivial, does that mean the solution is all real numbers or does it just mean that I used the wrong inequality, an inequality that doesn't solve the problem?

Thanks,

Tina Jin

 
WangDr. Kevin的头像
Re: MCIII Number Theory 3.12 and 3.13
WangDr. Kevin - 2023年12月29日 Friday 21:50
 

For 3.12:

It is sometimes easier if you let $n=\lfloor x\rfloor$.  Then the equation becomes $$x^2 - 4x + n = 0.$$  For the equation to have real solutions, the discriminant has to be nonnegative.  This gives a small range for $n$ (which is the integer part of $x$), and you can try each possible value of $n$ to solve for $x$.

For 3.13:

You already got a very important result: $x + 1/x = \lfloor{x}\rfloor + \lfloor 1/x \rfloor + 1$, which is an integer.  So what kind of integers can equal $x + 1/x$ for real number $x$? (In fact it can be any integer, positive or negative, except for $-1, 0, 1$.  You can prove this fact easily.)  Assume $x + 1/x = k$, where $k$ is an integer, and solve for $x$.  Then it will turn out that $k=\pm 2$ don't work either, but everything else works.  That solves part (1).  Use the result in Part (1) to prove part (2).

JinTina的头像
Re: MCIII Number Theory 3.12 and 3.13
JinTina - 2024年01月5日 Friday 14:44
 

Hello,

I still need some help on this question. I tested some values of k like 3, 4, 5, and -3, -4, -5, and they do work, but how do I prove it?

Thanks,

Tina Jin

WangDr. Kevin的头像
Re: MCIII Number Theory 3.12 and 3.13
WangDr. Kevin - 2024年01月6日 Saturday 00:53
 

You can use the quadratic formula to solve $x + 1/x = k$ in terms of $k$.  All integers satisfying $|k| \geq 3$ will work.  

JinTina的头像
Re: MCIII Number Theory 3.12 and 3.13
JinTina - 2024年01月14日 Sunday 10:54
 
I still need help on 3.13. I used the quadratic formula and the discriminant to get |k|≥2. How do you prove all |k|≥3 will work?