[seqfan] Re: Help with sequence
Ignacio Larrosa Cañestro
ilarrosa at mundo-r.com
Tue Aug 25 18:05:39 CEST 2009
Andrew Plewe wrote:
>> 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-
But thie equation can be posed as
n = y^2 - a^2x^2 = (y + ax)(y - ax)
and reduces to a system of linear equations
y + ax = d1
y - ax = d2
where d1*d2 = n, and easy to solve.
Saludos,
Ignacio Larrosa Cañestro
A Coruña (España)
ilarrosa at mundo-r.com
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