${x}^{2}+{y}^{2}=849-{x}^{2}{y}^{2}$

Find all possible $x$ values to the equation above such that $x$ and $y$ are both positive integers. And state the values of $x$ in ascending order with no spaces in between (e.g. if the solutions were $2$ , $55$ and $13$ you would write " $21355$ ").

The answer is 24713.

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Nice question and solution. One suggestion, though; I think you will need to add that $y$ must be an integer as well, for as presently stated we can find a real number $y$ for any given positive integer $x$ that will satisfy the equation. :)

Brian Charlesworth
- 6 years, 2 months ago

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If y is allowed to be a positive real number the number of solutions rises and x cqn be one of: $1,2,3,4,7,13$

So @Sophie add in the condition that y is a positive real number as well.

Sualeh Asif
- 6 years, 2 months ago

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That was my first attempt, but then I realized that for any positive integer $x$ we can find a positive real value $y$ that satisfies the equation. To obtain Sophie's posted solution, it will need to be specified that $y$ must be an integer.

Brian Charlesworth
- 6 years, 2 months ago

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@Brian Charlesworth – Thank you both - I have edited it and hope this clears up any ambiguity. I forgot to specify that $y$ must also be an integer!

Sophie Maclean
- 6 years, 2 months ago

@Brian Charlesworth – Actually I wasted two tries because too! Thankyou for clearing the ambiguity

Sualeh Asif
- 6 years, 2 months ago

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@Sualeh Asif – Sorry! I guess it's all very well thinking of a problem but you have to write down properly :) pleas let me know if there are any ambiguities

Sophie Maclean
- 6 years, 2 months ago

i just wrote an algorithm.. i hope its easy..

int main()
{
float n,m,i,j;
int flag;
for(i=0;i<30;i++){
flag=0;
n = (849-(i
*
i))/(1+(i
*
i));
m=sqrt(n);
j = (int)m;
if(m>0 && j==m)
flag=1;
if(flag==1)
printf("%d\n",(int)i );
}
return 0;
}

Shubham Agrawal
- 6 years, 2 months ago

Thanks for writing this up.

Bill Bell
- 6 years, 2 months ago

$x^2+y^2=849-x^2y^2$

$\Rightarrow x^2+y^2+x^2y^2=849 \\ \quad x^2y^2 + x^2+y^2+ 1=850 \\ \quad (x^2+1)(y^2+1) = 850$

We can use a spreadsheet to do the computation. Mine is as follows:

We note that $x=\{ 2,4,7,13\}$ . Therefore, the required answer is $\boxed{24713}$

4 Helpful
0 Interesting
0 Brilliant
0 Confused

In the first quadrant, the maximum values of $x$ and $y$ are both less than 30.

It's a simple matter to check the space of integer values of $x$ and $y$ for those that adhere to this relation.

How did the author of the problem do it?

PS: Incidentally, this way of providing for solutions of problems opens up a lot of possibilities!

1 Helpful
0 Interesting
0 Brilliant
0 Confused

I really like this method - not being able to code very well myself I am always impressed with answers like this. I have now also posted my own solution if you'd like to see an entirely different way of doing it.

Sophie Maclean
- 6 years, 2 months ago

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I've been writing code for some many years that I have difficulty thinking in any other way.

Bill Bell
- 6 years, 2 months ago

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Seeing as someone asked, this is my solution.

${x}^{2}+{y}^{2} = 849 - {x}^{2}{y}^{2}$

${x}^{2}{y}^{2} + {x}^{2}+{y}^{2} = 849$

${x}^{2}{y}^{2} + {x}^{2}+{y}^{2} + 1 = 850$

$({x}^{2}+1)({y}^{2}+1)=850$

The pairs of factors of 850 can now be considered.

$850= 1\times 850$

$850= 2\times 425$

$850= 5\times 170$

$850= 10\times 85$

$850= 17\times 50$

$850= 25\times 34$

It is clear that there are two occasions when both in the pair are one more than a square number so the possible solutions are:

${x}^{2}+1=5$ and ${y}^{2}+1=170$ (and vice versa)

or ${x}^{2}+1=17$ and ${y}^{2}+1=50$ (and vice versa)

Therefore $x=2$ or $13$ or $4$ or $7$ .

Of course, I had the advantage that I wrote the question for a friend with certain techniques in mind, though it is great to see other methods that I hadn't considered!