[seqfan] Re: near multiples of squares in Fibonacci, Lucas, Pell, etc. numbers

Georgi Guninski guninski at guninski.com
Wed Jun 5 11:47:02 CEST 2013


I suppose at most finitely many.

For x,y consecutive fibonacci numbers an identity is:
x^2+xy-y^2= \pm 1   [A]
where the choice of sign depends on whether x is F_{2n} or F_{2n+1}.
Set in [A] x= a u^2 + b u +c
In general [A] is genus 1 curve and has finitely many integral points.

In the rare case when it is genus 0, it might have infinitely many integral
points, so computing the genus solves finiteness.

On the other hand lucas numbers satisfy L_{2n} = L_n^2 -2 (-1)^n.

On Tue, Jun 04, 2013 at 09:16:49PM -0400, Allan Wechsler wrote:
> Are there _any_ quadratic polynomials that take on an infinite number of
> Fibonacci values for integral argument?
> 
> 
> On Tue, Jun 4, 2013 at 9:06 PM, Max Alekseyev <maxale at gmail.com> wrote:
> 
> > SeqFans,
> >
> > I've just published at arxiv a manuscript on finding integral points
> > on biquadratic curves with application to finding terms of the form
> > a*m^2 + b in Lucas sequences:
> > http://arxiv.org/abs/1306.0883
> > As an example I established that among Fibonacci numbers only 2 and 34
> > are of the form 2m^2+2; only 1, 13, and 1597 are of the form m^2-3;
> > and so on.
> >
> > I would appreciate any comments or suggestions.
> >
> > P.S. I'm going to present this result at CanaDAM 2013 conference next week.
> >
> > Regards,
> > Max
> >
> > _______________________________________________
> >
> > Seqfan Mailing list - http://list.seqfan.eu/
> >
> 
> _______________________________________________
> 
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