Re: trying to parameterize solutions for Pythagorean ratios and Diophantine m-tuples

[American Citizen <website.reader3@gmail.com>]

Thank you Bill. I was able to write up a general algorithm, using the 
general m,n input instead of m,n = 3,2 as shown in my example.

I also developed the forward/inverse maps, but discovered that my 
ellpool(e,p,ht) which takes a given curve e, a set of points p 
(hopefully the Mordell_Weil basis) and ht as the maximum height of the 
points found and added to the pool, is failing to give me all the points 
of 0 <= height(pt) <= ht on the elliptic curve e, so I have work to do 
now trying to discover why this pool algorithm is NOT finding those points.

But your answer gave me a way to directly find tons of points for a 
given m,n ratio, such that the p,q satisfy the a*b+1 = square condition. 
This is a big help!

Randall

On 3/3/24 04:41, Bill Allombert wrote:
> On Sat, Mar 02, 2024 at 05:22:15PM -0800, American Citizen wrote:
> > (3)    30*q*p^3 + 144*q^2*p^2 - 30*q^3*p = r^2 for p,q,r in Z
> >
> > This is almost an homogeneous quadratic equation in p,q made a square, but
> > the product of the powers = 3
> > for each binomial term on the left.
> >
> > Is there a parametric solution for (2) given m,n or (3) ?
> Divide 3) by q^4 and set x=p/q, y =r/q^2, you get
> y^2 = 30*x^3 + 144*x^2-30*x
>
> then set y = Y/30  x = X/30
>
> So
>
> Y^2 = X^3 + 144*X^2 - 900*X
>
> which is a rank-2 elliptic curve with full 2-torsion,
> so there are a lot of solutions.
>
> ? E=ellinit([0,144,0,-900,0])
> ? elltors(E)
> %8 = [4,[2,2],[[-150,0],[0,0]]]
> ? ellrank(E)
> %9 = [2,2,0,[[-90,720],[45,585]]]
>
> For example [-90,720] is [p,q] = [3,-1],
> and         [45,585]  is [p,q] = [3, 2]
>
> and ellmul(E,[-90,720],5)
> gives [-1724297003000010/567013106089369,-858234297395033146517040/13501737061684979934653]
> leads to [57476566766667, -567013106089369]
>
> Cheers,
> Bill.