Stein: Is it conjectured that the number of integral points on minimal curves is unbounded? If so, then abc implies that ranks are unbounded.

Villegas: If you say that the curve has rank  and look at regulator in terms of -function, then look at BSD, then big imply small regulator hence small height. If you believe that the -function is essentially constant....

Mazur: In rank 2 case, what is ratio between shortest and second shortest.

Weissman: Higher dimensional? S. David. Can formulate an analogue for abelian varieties, but I don't know much so I shouldn't summarize. See Number Theory III, by Lang.

Elkies: Thanks! The four examples you list are quite remarkable also for having lots of integer points: sixteen pairs on each of the top two examples (#1: 1-21 except 8,11,13,16,17; #2: 1-18 except 10,11,17 and also 21); fourteen on #3 (1-18 except 7,11,13,14), and fifteen on #4 (1-15 except 8,13, and also 18 and 25). In each case I don't claim that my list is complete--I looked only up to 50 and certainly didn't attempt a proof that there aren't yet more integer multiples, though for rank 1 it's reasonably straightforward to do in practice. Anyway, these examples beat the best I that I could find in the tables, where there were never more than 12 integral pairs, and the largest multiple was 24P for 735F. For consecutive integer points, the curve 618F ties your record of nine for 3630Y. I suspect that you might find even more such examples by setting a more generous limit on the height and looking specifically for many integral multiples, for integral with  large, or for many consecutive multiples (could there be a case where is integral for each in ?).

Update: is realized by the point on the curve of conductor . The point is also integral.

The largest integral multiple I can find for now is , for , (whose first multiples, and , , , are also integral).

Acknowledgement: It is a pleasure to thank Matt Baker for giving a really cool talk, the people who made comments above, and John Cremona for computing a table of regulators of elliptic curves.