Integer solutions for straight lines

[gortwell]

Integer solutions for hyperbola

Hi,

Is it possible to determine algebraically whether a hyperbola can have integer solutions for x and y?

i.e.

\(\displaystyle y^2=5x^2+20\) has integer solutions at

x	y
11	25
29	65
199	445

and many more (infinite ?)

However, I can't find any integer solutions for \(\displaystyle y^2=5x^2+45\)

Thanks

 

[pappus]

Hi,

Is it possible to determine algebraically whether a straight line can have integer solutions for x and y?

i.e.

\(\displaystyle y^2=5x^2+20\) has integer solutions at

x	y
11	25
29	65
199	445

and many more (infinite ?)

However, I can't find any integer solutions for \(\displaystyle y^2=5x^2+45\)

Thanks

I didn't use an algebraic method but brute force. I'm very interested to learn how you found the results of the other hyperbola.

(x, y)=(6, 15)

(x, y) = (114, 255)

 

[gortwell]

Hi Pappus,

I obtained the results for the first hyperbola (as you correctly state) by the same method as you

There's actually a constraint I forgot to include, x and y must be coprime.

It appears this is a form of Pell's equation, but still can't figure how to go about it

 

[gortwell]

others:
2046,4575 ..................both divisible by 3
35714,82095...............??? doesn't fit either equation
658806,1473135..........both divisible by 3

y^2 = 5x^2 + 45

x^2 = (y^2 - 45) / 5

x^2 = y^2/5 - 9 : so y divisible by 5, and x is even.

Hi Denis,

Maybe you missed my reply to pappus where I said that x and y must be coprime