[seqfan] Help with sequence

Andrew Plewe andrew at nevercenter.com
Tue Aug 25 17:38:48 CEST 2009

I've the distinct feeling that I've stepped into waters that are a bit  
above my pay grade.

Let a^2*x^2 + n = y^2, and let a and n vary over the set of positive  
integers. If I understand the Hasse-Minkowski Theorem correctly, I can  
solve that equation mod some integer p to demonstrate that solutions  
for it exist (or don't). I believe this is what Dario Alpern's  
Diophantine Quadratic Equation Solver does in its first few steps  
(when it checks the equation mod 9, 16, and 25 to see if it has  
solutions). Anyways, the idea I have is to construct a table of the  
minimum values of p necessary to demonstrate that my equation has or  
doesn't have solutions for all combinations of a and n.

I've read elsewhere that it's possible to compute a single value for p  
that is sufficient to demonstrate if the equation has solutions, but  
the literature is a bit too dense for me to figure out how to do it,  
or if that value is the smallest value of p which demonstrates that  
the equation has solutions. I'd appreciate any help in understanding  
how this works. Thanks!

     -Andrew Plewe-

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