MHB Solving for x and k: Positive Ints

  • Thread starter Thread starter Albert1
  • Start date Start date
  • Tags Tags
    Positive
AI Thread Summary
The discussion focuses on finding positive integers x and k that satisfy the equation x(x+1) = 64k^4 - 48k^3 + 16k^2 - 2k. The right-hand side can be factorized, leading to the discovery that for k=1, x equals 5. Participants argue that for k greater than 1, x does not exist, as the factors would not yield consecutive integers. The conversation also touches on the nature of factorization, distinguishing between algebraic and arithmetic forms, and emphasizes that the only viable solution found is k=1 and x=5. The conclusion suggests that while other solutions may seem unlikely, they cannot be entirely ruled out without further exploration.
Albert1
Messages
1,221
Reaction score
0
x and k are positive integers ,satisfying :

$ \text {x(x+1)=64} \text { k}^4 -48 \text{ k}^3 +16\text { k}^2-2\text{ k} $

please find x
 
Mathematics news on Phys.org
Albert said:
x and k are positive integers ,satisfying :

$ \text {x(x+1)=64} \text { k}^4 -48 \text{ k}^3 +16\text { k}^2-2\text{ k} $

please find x

Your RHS can be factorized as $2k(4k-1)(8k^2-4k+1)$.
Trying it out for k=1, we find:
$2k(4k-1)(8k^2-4k+1) = 2 \cdot 3 \cdot 5 = 5 \cdot 6$​

So I've found $x=5$. $\qquad \blacksquare$

Found it! Phew! (Whew)
 
ILikeSerena: you got the answer (Yes)

now I think you should point out if k>1 then x will not exist

what do you think ?
 
$2k(4k-1)(8k^2-4k+1)=(8k^2-2k)(8k^2-4k+1) $

$\text{if} \,\, 8k^2-4k+1 < 8k^2-2k , \,\text{then}\, k>\dfrac {1}{2} $

let :

$ x=8k^2-4k+1 , \, x+1=8k^2-2k $

we have :

$ 8k^2-4k+2=8k^2-2k $

$\therefore k=1 , \, \, x=5 $
 
I don't get it.
How do you know that x does not exist for k > 1?
 
ILikeSerena said:
I don't get it.
How do you know that x does not exist for k > 1?
please take note of my previous post as follows :
Albert said:
[FONT=MathJax_Main]2[FONT=MathJax_Math]k[FONT=MathJax_Main]([FONT=MathJax_Main]4[FONT=MathJax_Math]k[FONT=MathJax_Main]−[FONT=MathJax_Main]1[FONT=MathJax_Main])[FONT=MathJax_Main]([FONT=MathJax_Main]8[FONT=MathJax_Math]k[FONT=MathJax_Main]2[FONT=MathJax_Main]−[FONT=MathJax_Main]4[FONT=MathJax_Math]k[FONT=MathJax_Main]+[FONT=MathJax_Main]1[FONT=MathJax_Main])[FONT=MathJax_Main]=[FONT=MathJax_Main]([FONT=MathJax_Main]8[FONT=MathJax_Math]k[FONT=MathJax_Main]2[FONT=MathJax_Main]−[FONT=MathJax_Main]2[FONT=MathJax_Math]k[FONT=MathJax_Main])[FONT=MathJax_Main]([FONT=MathJax_Main]8[FONT=MathJax_Math]k[FONT=MathJax_Main]2[FONT=MathJax_Main]−[FONT=MathJax_Main]4[FONT=MathJax_Math]k[FONT=MathJax_Main]+[FONT=MathJax_Main]1[FONT=MathJax_Main])

[FONT=MathJax_Main]if[FONT=MathJax_Main] 8[FONT=MathJax_Math]k[FONT=MathJax_Main]2[FONT=MathJax_Main]−[FONT=MathJax_Main]4[FONT=MathJax_Math]k[FONT=MathJax_Main]+[FONT=MathJax_Main]1[FONT=MathJax_Main]<[FONT=MathJax_Main]8[FONT=MathJax_Math]k[FONT=MathJax_Main]2[FONT=MathJax_Main]−[FONT=MathJax_Main]2[FONT=MathJax_Math]k[FONT=MathJax_Main],[FONT=MathJax_Main]then [FONT=MathJax_Math]k[FONT=MathJax_Main]>[FONT=MathJax_Main]1/2

let :

[FONT=MathJax_Math]x[FONT=MathJax_Main]=[FONT=MathJax_Main]8[FONT=MathJax_Math]k[FONT=MathJax_Main]2[FONT=MathJax_Main]−[FONT=MathJax_Main]4[FONT=MathJax_Math]k[FONT=MathJax_Main]+[FONT=MathJax_Main]1[FONT=MathJax_Main],[FONT=MathJax_Math]x[FONT=MathJax_Main]+[FONT=MathJax_Main]1[FONT=MathJax_Main]=[FONT=MathJax_Main]8[FONT=MathJax_Math]k[FONT=MathJax_Main]2[FONT=MathJax_Main]−[FONT=MathJax_Main]2[FONT=MathJax_Math]k

we have :

[FONT=MathJax_Main]8[FONT=MathJax_Math]k[FONT=MathJax_Main]2[FONT=MathJax_Main]−[FONT=MathJax_Main]4[FONT=MathJax_Math]k[FONT=MathJax_Main]+[FONT=MathJax_Main]2[FONT=MathJax_Main]=[FONT=MathJax_Main]8[FONT=MathJax_Math]k[FONT=MathJax_Main]2[FONT=MathJax_Main]−[FONT=MathJax_Main]2[FONT=MathJax_Math]k

[FONT=MathJax_AMS]∴[FONT=MathJax_Math]k[FONT=MathJax_Main]=[FONT=MathJax_Main]1[FONT=MathJax_Main],[FONT=MathJax_Math]x[FONT=MathJax_Main]=[FONT=MathJax_Main]5

$ 8k^2-4k+2=8k^2-2k -------------------(*) $

the only value for k satisfying (*) is k=1
 
Albert said:
please take note of my previous post as follows :$ 8k^2-4k+2=8k^2-2k -------------------(*) $

the only value for k satisfying (*) is k=1

But how did you get $x=8k^2−4k+1$ then?
Isn't that just a try-out?
 
Albert said:
$2k(4k-1)(8k^2-4k+1)=(8k^2-2k)(8k^2-4k+1) $

$\text{if} \,\, 8k^2-4k+1 < 8k^2-2k , \,\text{then}\, k>\dfrac {1}{2} $

let :

$ x=8k^2-4k+1 , \, x+1=8k^2-2k $

we have :

$ 8k^2-4k+2=8k^2-2k $

$\therefore k=1 , \, \, x=5 $
What this shows is that if the integer $N(k) \overset{\text{def}}{=} 64k^4 - 48k^3 + 16k^2 -2k$ is factorised as the product of the two factors $8k^2-2k$ and $8k^2-4k+1$, then the only way for those factors to differ by $1$ is if $k=1$. That still leaves open the possibility that $N(k)$ could be factorised in some different way as the product of two integers differing by $1$. So I agree with ILikeSerena that the above argument does not rule out the possibility of other solutions (though admittedly it seems unlikely that there are any).
 
Opalg said:
So I agree with ILikeSerena that the above argument does not rule out the possibility of other solutions (though admittedly it seems unlikely that there are any).

I checked up to k=100000 with N(100000) = 6399952000159999800000.
No other solutions though.
 
  • #10
as you said my RHS can be factorized as :
$2k(4k-1)(8k^2-4k+1)=(8k^2-2k)(8k^2-4k+1)=(x+1)x$

one of the two factors (the smaller one ) must =x
and the other factor (bigger) must =x+1 (they are two consecutive positive integers)
and we can judge :
$ 8k^2-4k+1 (smaller)<8k^2-2k (bigger)$
so we set $ x=8k^2-4k+1 $
 
  • #11
Albert said:
as you said my RHS can be factorized as :
$2k(4k-1)(8k^2-4k+1)=(8k^2-2k)(8k^2-4k+1)=(x+1)x$

one of the two factors (the smaller one ) must =x
and the other factor (bigger) must =x+1 (they are two consecutive positive integers)
and we can judge :
$ 8k^2-4k+1 (smaller)<8k^2-2k (bigger)$
so we set $ x=8k^2-4k+1 $

That would only hold if one of those factors is a prime number.
In general neither is.
 
  • #12
Albert said:
as you said my RHS can be factorized as :
$2k(4k-1)(8k^2-4k+1)=(8k^2-2k)(8k^2-4k+1)=(x+1)x$

one of the two factors (the smaller one ) must =x
and the other factor (bigger) must =x+1 (they are two consecutive positive integers)
and we can judge :
$ 8k^2-4k+1 (smaller)<8k^2-2k (bigger)$
so we set $ x=8k^2-4k+1 $
I think that you are confusing algebraic factorisation with arithmetic factorisation. To take a simple example, if $f(x) = x^2+6x+5$ then $f(x)$ has only one algebraic factorisation, namely $f(x) = (x+1)(x+5)$. If you put $x=1$, then $f(1) = 12$ and the two factors $x+1$ and $x+5$ give you the factorisation $f(1) = 2\times 6$. But if you consider it arithmetically, as a number, then $12$ has other factorisations, such as $12 = 3\times 4$, which are not apparent from the algebraic factorisation of $f(x)$.

For the problem in this thread, we are looking for arithmetic factorisations, and these are much more elusive than the algebraic factorisation suggests.
 
  • #13
if we set :
$x+1=8k^2-4k+1 , x=8k^2-2k$
then k=0
this does not fit ,because x and k are positive integers

and you said :"That would only hold if one of those factors is a prime number.
In general neither is."

x or x+1 in this case one of them is a prime ,but in other case may be both of them are composite numbers ( for y=8, and y+1=9),anyway they must be coprime

that is :
$ 8k^2-4k+1 , and \,\, 8k^2-2k $ are coprime

using this trait we can find the corresponding value of k and x
 
Last edited:

Similar threads

Replies
8
Views
2K
Replies
2
Views
2K
Replies
7
Views
2K
Replies
3
Views
1K
Replies
3
Views
1K
Replies
2
Views
950
Replies
3
Views
2K
Back
Top