Let's assume that $d>1$ is a squarefree integer. If I am given an integer $m$, is there a way to use algebraic number theory to determine whether or not $x^2-dy^2=m$ has a solution in integers? For example, if $\mathbb Q(\sqrt d)$ has class number $1$, then I simply need to check to see if there is an ideal in the ring of integers of that quadratic extension whose norm is $\pm m$ (and then check the norms of units to see if I can get the correct sign). If $\mathbb Q(\sqrt d)$ doesn't have class number $1$, I can't use the fact that norms of elements are just norms of ideals (up to signs). Is there still a way I can proceed in this case?
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1https://math.stackexchange.com/questions/2653260/existence-of-solution-to-generalized-pells-equation – saulspatz Apr 12 '18 at 23:48
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Above equation shown below:
$x^2-dy^2=m$
In the above equation if "m' is a square number then the above equation always has solutions in integer's given below in parametric form (Reference: History of number theory by L.E. Dickson)
$(x,y,z,d) = ((v^2-2u),(2v),(v^4-4u^2+4uv^2),(u^2-uv^2))$
Sam
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In my previous answer $(x,y,w)$ are all square numbers.The equation
$(x^2-dy^2=m)$ is then equivalent to
$(p^4-dq^4=w^2)$
Sam
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