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# Superbrain Q watch

1. Find all integers x and y which satisfy the equation
x² - 3y² = 17

taken from
http://www.ucc.ie/mathsoc/pastpapers/s1987.shtml

I cannot figure out how to start. Never dealt with anything of this type before.
2. Try to solve it mod 3.
3. Google Pell equations and continued fractions.
4. (Original post by Mr M)
Google Pell equations and continued fractions.
Thanks
5. (Original post by Mr M)
Google Pell equations and continued fractions.
Unless I'm mistaken, that's not the best advice here.
6. (Original post by DFranklin)
Unless I'm mistaken, that's not the best advice here.
He said he had never dealt with equations of this type so I was trying to help him research the topic further. I had already seen your post.

Message to mrmanps: In matters such as this, it is probably best to listen to DF first and me second.
7. (Original post by DFranklin)
Try to solve it mod 3.
Is there a good website you could point me to in order to recap this? It's been a while
8. http://en.wikipedia.org/wiki/Modular_arithmetic ?

(In this particular case, reducing mod 3 eliminates y, and you can then use trial and error to find all possible solutions in x).
9. (Original post by DFranklin)
http://en.wikipedia.org/wiki/Modular_arithmetic ?

(In this particular case, reducing mod 3 eliminates y, and you can then use trial and error to find all possible solutions in x).
so it becomes x²=2?
10. (Original post by mrmanps)
so it becomes x²=2?
x^2 = 2 mod 3, yeah.
11. (Original post by around)
x^2 = 2 mod 3, yeah.
so how am i suppose to find an integer vale of x now?
12. (Original post by mrmanps)
so how am i suppose to find an integer vale of x now?
Are you certain there are integer solutions to find?
13. (Original post by Mr M)
Are you certain there are integer solutions to find?
Q10 from this past paper
taken from
http://www.ucc.ie/mathsoc/pastpapers/s1987.shtml
14. Maybe substituting x=3n+2 would help? It may not though, just an idea.
15. (Original post by james22)
Maybe substituting x=3n+2 would help? It may not though, just an idea.
reduces to 9n²+12n+2=0 which doesn't have integer roots
16. (Original post by mrmanps)
...
Can you prove there are no square numbers that leave a remainder of 2 after division by 3?
17. I've just managed to deduce that there are no integer solutions. though my method is a bit rough and probobly has a few errors in it.

I used the fact that x^2 must be of the form 3n+2, then made that substitution into the original equation, solved it for n and showed that there is no integer n which makes y an integer. It involved showing that 3y^2+19 colud never be a square which was the hardest part.
18. (Original post by james22)
I've just managed to deduce that there are no integer solutions. though my method is a bit rough and probobly has a few errors in it.

I used the fact that x^2 must be of the form 3n+2, then made that substitution into the original equation, solved it for n and showed that there is no integer n which makes y an integer. It involved showing that 3y^2+19 colud never be a square which was the hardest part.
So this question (taken from an exam paper) see the first thread, is not possible to find a pair of integer solutions for? odd
19. (Original post by mrmanps)
So this question (taken from an exam paper) see the first thread, is not possible to find a pair of integer solutions for? odd
Well a solution can be that there are no solutions.
20. (Original post by Mr M)
Well a solution can be that there are no solutions.
True.

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