Superbrain Q

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.

Reply 1

DFranklin

18

Try to solve it mod 3.

Google Pell equations and continued fractions.

Reply 3

mrmanps

OP

11

Original post by Mr M

Google Pell equations and continued fractions.

Thanks

Reply 4

DFranklin

18

Original post by Mr M

Google Pell equations and continued fractions.

Unless I'm mistaken, that's not the best advice here.

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.

(edited 12 years ago)

Reply 6

mrmanps

OP

11

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

Reply 7

DFranklin

18

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).

Reply 8

mrmanps

OP

11

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?

Reply 9

around

13

Original post by mrmanps

so it becomes x²=2?

x^2 = 2 mod 3, yeah.

Reply 10

mrmanps

OP

11

Original post by around

x^2 = 2 mod 3, yeah.

so how am i suppose to find an integer vale of x now?

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?

Reply 12

mrmanps

OP

11

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

Reply 13

james22

16

Maybe substituting x=3n+2 would help? It may not though, just an idea.

Reply 14

mrmanps

OP

11

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

Original post by mrmanps

...

Can you prove there are no square numbers that leave a remainder of 2 after division by 3?

Reply 16

james22

16

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.

(edited 12 years ago)

Reply 17

mrmanps

OP

11

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.