[seqfan] Help with sequence
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!
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