One binary form.
by individ, Jan 17, 2015, 7:41 AM
For such equations:
![\[\frac{x^2+y^2}{xy-1}=-t^2\]](//latex.artofproblemsolving.com/a/5/b/a5b4482f1d6dfa04d8ac4d6dbb3f40ef7102cfac.png)
Using the solutions of the Pell equation.![\[p^2-(t^4-4)s^2=1\]](//latex.artofproblemsolving.com/6/3/d/63d47b43cf385512660987adb4eb28734c1a7b22.png)
You can write the solution.
![\[x=-4tps\]](//latex.artofproblemsolving.com/1/d/d/1dd6d59fa4ae2621924a4f4ed2f185f8c34e7bee.png)
![\[y=t(p^2+2t^2ps+(t^4-4)s^2)\]](//latex.artofproblemsolving.com/2/a/4/2a48bedc80183e2cc51df926b145d4e2efc769e0.png)
It all comes down to the Pell equation - as I said.
Considering specifically the equation:
![\[\frac{x^2+y^2}{xy-1}=5\]](//latex.artofproblemsolving.com/f/0/a/f0ab670373fb61f302873bd8cb13d00ab428f992.png)
Decisions are determined such consistency. Where the next value is determined using the previous one.
![\[p_2=55p_1+252s_1\]](//latex.artofproblemsolving.com/b/2/3/b23bc3157452eeaa0cda9319626a3dfd45b2e8c2.png)
![\[s_2=12p_1+55s_1\]](//latex.artofproblemsolving.com/1/7/a/17a8c18f63378231f4ff5a330eeb5ef88e26221b.png)
You start with numbers.
Using these numbers, the solution can be written according to a formula.
![\[y=p^2+2ps+21s^2\]](//latex.artofproblemsolving.com/5/0/8/508336db9503829afca688304ee48369c27dea2e.png)
![\[x=3p^2+26ps+63s^2\]](//latex.artofproblemsolving.com/8/6/5/8656026f32cfaaf7d8882428aa7dbbbad276b721.png)
If you use an initial
Then the solutions are and are determined by formula.
![\[y=s\]](//latex.artofproblemsolving.com/d/9/c/d9c7fc6dc0b76dda2eafeefd839af034ba9078b0.png)
![\[x=\frac{p+5s}{2}\]](//latex.artofproblemsolving.com/8/f/2/8f28b402baafb57e17717c3a35d40106af2c9b1a.png)
As the sequence it is possible to write endlessly. Then the solutions of the equation, too, can be infinite.
If you use a sequence with the first element.
- 
Then decisions can be recorded.
![\[y=2s\]](//latex.artofproblemsolving.com/9/f/f/9ff757d99738ecd89ccc1fcc8155b8af99acc68e.png)
![\[x=p+5s\]](//latex.artofproblemsolving.com/9/3/4/934e7fa1b9db4e9fed82986102e16a5570496d01.png)
If you use a sequence with the first element.
- 
Using this sequence can be different. On its basis with the first element.
- 
![\[z_2=pz_1+7sq_1\]](//latex.artofproblemsolving.com/e/1/1/e11a422cd330b3afe991e0692580e1e8f0f28f5d.png)
![\[q_2=pq_1+3sz_1\]](//latex.artofproblemsolving.com/9/5/c/95c754237bd2e153084ef035a2eb3f6508c44de3.png)
Then decisions will be.
![\[x=z-q\]](//latex.artofproblemsolving.com/6/6/4/6642fd4b5dc9126fa73a4e90923da04f6ade4215.png)
![\[y=z+q\]](//latex.artofproblemsolving.com/f/d/8/fd8657dee72104f2054716487033f94f6aa17656.png)
It is necessary to take into account that the number can have a different sign. -
![\[\frac{x^2+y^2}{xy-1}=-t^2\]](http://latex.artofproblemsolving.com/a/5/b/a5b4482f1d6dfa04d8ac4d6dbb3f40ef7102cfac.png)
Using the solutions of the Pell equation.
![\[p^2-(t^4-4)s^2=1\]](http://latex.artofproblemsolving.com/6/3/d/63d47b43cf385512660987adb4eb28734c1a7b22.png)
You can write the solution.
![\[x=-4tps\]](http://latex.artofproblemsolving.com/1/d/d/1dd6d59fa4ae2621924a4f4ed2f185f8c34e7bee.png)
![\[y=t(p^2+2t^2ps+(t^4-4)s^2)\]](http://latex.artofproblemsolving.com/2/a/4/2a48bedc80183e2cc51df926b145d4e2efc769e0.png)
It all comes down to the Pell equation - as I said.
Considering specifically the equation:
![\[\frac{x^2+y^2}{xy-1}=5\]](http://latex.artofproblemsolving.com/f/0/a/f0ab670373fb61f302873bd8cb13d00ab428f992.png)
Decisions are determined such consistency. Where the next value is determined using the previous one.
![\[p_2=55p_1+252s_1\]](http://latex.artofproblemsolving.com/b/2/3/b23bc3157452eeaa0cda9319626a3dfd45b2e8c2.png)
![\[s_2=12p_1+55s_1\]](http://latex.artofproblemsolving.com/1/7/a/17a8c18f63378231f4ff5a330eeb5ef88e26221b.png)
You start with numbers.
Using these numbers, the solution can be written according to a formula.
![\[y=p^2+2ps+21s^2\]](http://latex.artofproblemsolving.com/5/0/8/508336db9503829afca688304ee48369c27dea2e.png)
![\[x=3p^2+26ps+63s^2\]](http://latex.artofproblemsolving.com/8/6/5/8656026f32cfaaf7d8882428aa7dbbbad276b721.png)
If you use an initial
Then the solutions are and are determined by formula.
![\[y=s\]](http://latex.artofproblemsolving.com/d/9/c/d9c7fc6dc0b76dda2eafeefd839af034ba9078b0.png)
![\[x=\frac{p+5s}{2}\]](http://latex.artofproblemsolving.com/8/f/2/8f28b402baafb57e17717c3a35d40106af2c9b1a.png)
As the sequence it is possible to write endlessly. Then the solutions of the equation, too, can be infinite.
If you use a sequence with the first element.
- 
Then decisions can be recorded.
![\[y=2s\]](http://latex.artofproblemsolving.com/9/f/f/9ff757d99738ecd89ccc1fcc8155b8af99acc68e.png)
![\[x=p+5s\]](http://latex.artofproblemsolving.com/9/3/4/934e7fa1b9db4e9fed82986102e16a5570496d01.png)
If you use a sequence with the first element.
- 
Using this sequence can be different. On its basis with the first element.
- 
![\[z_2=pz_1+7sq_1\]](http://latex.artofproblemsolving.com/e/1/1/e11a422cd330b3afe991e0692580e1e8f0f28f5d.png)
![\[q_2=pq_1+3sz_1\]](http://latex.artofproblemsolving.com/9/5/c/95c754237bd2e153084ef035a2eb3f6508c44de3.png)
Then decisions will be.
![\[x=z-q\]](http://latex.artofproblemsolving.com/6/6/4/6642fd4b5dc9126fa73a4e90923da04f6ade4215.png)
![\[y=z+q\]](http://latex.artofproblemsolving.com/f/d/8/fd8657dee72104f2054716487033f94f6aa17656.png)
It is necessary to take into account that the number can have a different sign. -
This post has been edited 2 times. Last edited by individ, Feb 21, 2015, 5:42 AM
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