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

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

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