
Re: Pell type equations
Posted:
Apr 28, 2012 8:39 AM


Terry M wrote:
> The reason I was hoping there was a way to plug this type of equation into > Brahmagupta's method of composition is that I can find many solutions for > > 5x^2 + 20 = y^2 > > where x and y are coprime but can find no such solutions for 5x^2 + 45 = > y^2
As before, 5  y, and writing y = 5z we get
x^2  5z^2 = 9.
An equation of the form x^2  dy^2 = c (where d is not a perfect square) may have no solution; but if you can find one you can find an infinite number by combining this solution with the general solution of Pell's equation x^2  dy^2 = 1 (which always has an infinity of solutions), in the way I suggested.
The equation x^2  dy^2 = c has a solution if it has a solution modulo 8d, I think.
In your case this means there must be solutions mod 8 and mod 5, which there are.
Actually, in this case it is sufficient to find a solution of
u^2  5v^2 = 1
since then x = 3u, z = 3v (and it's not difficult to see that every solution must be of this form, ie x and z must be divisible by 3).
This has the trivial solution (u,v) = (2,1), or (x,z) = (6,3) or (x,y) = (6,15)
So there are an infinity of solutions, which you can get in the way I suggested. Eg if e = 2 + sqrt5 then e^3 = 38 + 17 sqrt5, so (u,v) = (38,17) is a solution of u^2  5y^2 = 1, giving (3 x 38, 15 x 17) as a solution of the original equation.
 Timothy Murphy email: gayleard /at/ eircom.net tel: +353862336090, +35312842366 smail: School of Mathematics, Trinity College Dublin

