American Citizen on Mon, 04 Mar 2024 00:38:11 +0100


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


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.