Re: x^2 - Ay^2 =1

Vincenzo Librandi wrote :
Philippe wrote:

We are waiting for your polynomials giving
A = 61, 109 or 421...

1766319049^2-61*226153980^2=1
is tied

Hi,

I don't understand what you mean by 'tied'
tied to what ? related with the =-1 equation ?

Also, yes you have a solution to that specific equation, but
not with your method of polynomial representation for A !
(61 is not in the list of your allready given polynomial forms)

29718^2-61*3805^2=-1
find the tie.

what tie ???
if X, Y is a solution of any generalized Pell's equation
x^2 - A*y^2 = -1, then
X^2 + A*Y^2, 2X*Y is a solution of
x^2 - A*y^2 = +1

This is true for all A for which x^2 - A*y^2 = -1 has a solution.

And generalized into :
If X,Y is solution of x^2 - A*y^2 = p
and U,V solution of x^2 - A*y^2 = q, then
U*X + A*V*Y, U*Y + V*X is solution of x^2 - A*y^2 = p*q

For others: tied x^2-109*Y^2=-1
or x^2-421y^1=-1, and I fiend what ask.

Again, I don't understand your 'tied'.
None of 109 and 421 is in your list of polynomials.
What do you mean by 'fiend' ?

And as above, if we know a solution of x^2 - A*y^2 = -1,
then we get immediately the solution for x^2 - A*y^2 = +1.
So *if* x^2-109*Y^2=-1 or x^2-421y^1=-1 have solutions (and they do)
we can deduce from these solutions the solution of the +1 equations.

But if the equation x^2 - A*y^2 = -1 has no solutions, and many
of them don't, we can deduce nothing about solution of the
corresponding x^2 - A*y^2 = +1

Also the problem is not in just solving these equations, but
to apply _your_ method in solving them.
That is find polynomial forms for A, X, Y which fit to this equation.




I have reordered your answers, the two equations are independent.

I give you a solution, which is may be or not the fundamental one,
of this equation : x^2 - 21*y^2 = 1
X = 665335, Y = 145188
Could you say if this solution is fundamental or not ?
Could you find from this the fundamental solution ?
show me your 'simple decomposition'...

Of course [...]
665335=(5,11,12097) isn't square
so 55=(5,11)isn't square; and 55^2-21*12^2=1.


Usually just write 665335 = 5*11*12097 for prime decompositions,
writing (a,b,c...) is confusing.

I find the metod !

What method ?
You fell into a lucky coincidence for the 55.

I would have given
X = 73180801, Y = 15969360 instead of the 665335, what would
your "method" have given ?
73180801 = 17*31*138863, yes it is not a square, neither are
17*31 = 527
nor 17*138863 = 2360671
nor 31*138863 = 4304753
neither of these three being solutions of x^2 - 21y^2 = 1
although 73180801 _is_ a solution, and is not the fundamental
solution. What do you deduce ???
The same if I had given X = 6049 = 23*263.

Yes 55 is the fundamental solution, that you didn't proove, although
with y = 12 a proof could be just try and check all values
y = 1, 2... 11, or just solve the Pell's equation using classical
methods, what you reject.

And all solutions of this equation (A=21) are, in increasing order :

x = 1, y = 0 (the trivial solution of all Pell's equations)
x = 55, y = 12 (the fundamental solution of this equation)
x = 6049, y = 1320
x = 665335, y = 145188
x = 73180801, y = 15969360
x = 8049222775, y = 1756484412
x = 885341324449, y = 193197315960
etc... (from allready discussed recurrence relations)
and there are _no_ others, just because we start from the
fundamental one, apart changing x into -x and/or y into -y.

[reordered, this is the second equation, unrelated with the first]

Another example
x^2 - 85*y^2 = 1
X = 285769, Y = 30996
same question.

Of course [... and] 285769 = (11,83,313) isn't square

Do you mean that 913 = 11*83 should be a solution of
x^2 - 85*y^2 = 1 ? or just 11, or 83 ... ?
Did you proove that 285769 is or not the fundamental solution for
this equation ?

A _method_ is supposed to work in all cases, or should allways
be able to distinguish between cases where it applies and cases
where it doesn't !

Regards.

--
Philippe C., mail : chephip+news@xxxxxxx
site : http://chephip.free.fr/ (recreational mathematics)


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