American Citizen on Mon, 04 Mar 2024 02:18:26 +0100


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


To all:

Some one stated that all rational points on an elliptic curve can be inductively generated, if we have a Mordell-Weil basis (or subset)

They state (for a rank 2 curve) with Mordell-Weil basis P and Q

that all rational points are a composition of

{ uP + vQ for u,v in Z }

Does this mean that some weird combination of 1000000 * P + 938471*Q might produce a point of low height?

I naively started with a given P,Q, found all possible points of P+Q, P-Q, shoved it into a pool, restarted and for any pair Pi and Qi, I found the addition point and subtraction point and if they were less than the given height, added them into the pool. I kept doing this until no more points could be added to the pool. I just now found out that P-Q is NOT the same as Q-P, and have to fix my pool algorithm.

Unfortunately I am now finding out that this naive algorithm is missing rational points < given height.

If the uP + vQ composition law is true, will I be tripped up by huge values for u,v ??? (going to a small height point)

Randall