1. ## Diophantine equation .

Hi everyone !
Solve in N˛ this diophantine equation :
$y^2=x^3-5x$
Good luck !

Hi everyone !
Solve in N˛ this diophantine equation :
$y^2=x^3-5x$
Good luck !
Try to draw this function...

You will see that you need to check only values of x between -2 and 2.

3. Hi !
I actually do have a solution using graphs , equation of tangents , but it is too long and of course not beautiful at all !
I wanted a more "algebraic" solution , and i started this way :
if x=0 then y=0
suppose that x and y are different from 0
If x<0
$y^2=x^3-5x$ $\Leftrightarrow y^2=x(x^2-5)$ since y˛ is positif and x is negatif x˛-5 must be negatif and we get $x\in \left \{-2;-1 \right \}$
if x=-2 we get y˛=2 which is impossible , and if x=-1 we get y=2 or y=-2
so (-1,2) and (-1,-2) are solutions.
We now have to treat the case of x>0 , and here is where i got stuck !

4. Originally Posted by Also sprach Zarathustra
Try to draw this function...

You will see that you need to check only values of x between -2 and 2.

I can't see why: $(5,10)\,,\,(5,-10)$ are solutions, say...

Tonio

5. Originally Posted by tonio
I can't see why: $(5,10)\,,\,(5,-10)$ are solutions, say...

Tonio

Me too...

6. Hi again !
thanks for paying attention to my exercice !
Here is what Wolfram gave me
it also gave me that the only integer solutions are the ones i found , really weird !!!! I think we should use the gcd !

7. For $x,y > 0$ :

We have that $x\cdot (x^2 - 5)$ is a square, now of course $\text{gcd}(x,x^2-5)$ divides $x\cdot x + (-1)\cdot (x^2 - 5) = 5$ thus $\text{gcd}(x,x^2-5)$ is either 1 or 5 (in case that $5|x$ )

So we separate in the 2 cases:

Case 1: $\text{gcd}(x,x^2-5) = 1$.

Here it immediately follows that both $x$ and $x^2-5$ should be squares for their product to be a square.

However, note that $x^2-5$ if $x$ is big enough (since the "distance" between 2 consecutive squares increases) it can't be an square since $x^2$ is already an square. How big is big enough? note that for $x>3$ it will be impossible, and in fact you can see we get no solutions for this case.

Case 2: $\text{gcd}(x,x^2-5) = 5$.

Here $x = 5\cdot k$ thus $y^2 = 5^2\cdot k \cdot (5\cdot k^2 - 1)$

Thus $\left(\displaystyle\frac{y}{5}\right)^2 = k \cdot (5\cdot k^2 - 1)$ but now of course $k$ and $5\cdot k^2 - 1$ are coprime and so they both must be squares.

If $k=1$ we have a solution ( $x = 5$ ; $y=10$ ).

$k = a^2$ then we want $5\cdot a^4 -1 = b^2$

mmmm... stuck.

8. An Idea...

We need to prove that there are only solutions of the form: (a,2a) {and (a,-2a)}.

...and if we prove that we can make a conclusion that the only solution are: (0,0),(-1,2),(-1,-2),(5,10),(5,-10).
{by solving: a^3-4a^2-5a=0}

9. Originally Posted by Also sprach Zarathustra
An Idea...

We need to prove that there are only solutions of the form: (a,2a) {and (a,-2a)}.

...and if we prove that we can make a conclusion that the only solution are: (0,0),(-1,2),(-1,-2),(5,10),(5,-10).
{by solving: a^3-4a^2-5a=0}
Using parity may be ?

10. By showing that $\sqrt{2n(4n^2-5)}$ is not an integer we will conclude that $a$ is must to be odd.

e.i:
...hence(after proving the above) $a$ is odd, say $a=2n+1$ it easy to show that $y$ as integer must be even:
$y=\sqrt{(2n+1)^3-5(2n+1)}=\sqrt{4(2n+1)(n^2+n-1)}=2\sqrt{(2n+1)(n^2+n-1)}$

11. Originally Posted by Also sprach Zarathustra
By showing that $\sqrt{2n(4n^2-5)}$ is not an integer we will conclude that $a$ is must to be odd.
Well if we consider the functions f such that $f(x)=8x^3-10x$ and g such that $g(x)=x^2$ we find that their graphs never meet if i'm not wrong of course ....

Well if we consider the functions f such that $f(x)=8x^3-10x$ and g such that $g(x)=x^2$ we find that their graphs never meet if i'm not wrong of course ....
Interesting!

13. Originally Posted by Also sprach Zarathustra
Interesting!
Thanks !
By the way , as i said , i have a solution using graphs , it is similar to what we did !
It's in french , if you are interested in it , i'll publish the file so that you can take a look at it

Thanks !
By the way , as i said , i have a solution using graphs , it is similar to what we did !
It's in french , if you are interested in it , i'll publish the file so that you can take a look at it
I will be glad to look at it.
My French is like your Hebrew... Merci mon ami.

15. Originally Posted by Also sprach Zarathustra
I will be glad to look at it.
My French is like your Hebrew... Merci mon ami.
Your french is good apparently , but for me , i don't know a single word of Hebrew
Here is the file http://www.animath.fr/IMG/pdf/cours-arith1.pdf see page 68-.. (i hope that the mods and admins don't mind )

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