We give asymptotics for sums of sequences over smooth indices. In particular we obtain bounds for exponential sums over smooth numbers, and therefore estimates for the number of smooth triples \( a, b, a+b\) up to \( x\)

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Some years ago I presented a heuristic that, in the family of quadratic twists of a given elliptic curve, the rank is absolutely bounded, the proposed bound depending only on the number of rational 2-torsion points. At the time this contradicted the popular belief. Mark Watkins took it upon himself to do a massive calculation of ranks of quadratic twists of the congruent number curve, to test out my "conjecture". This paper is the record of an enormous calculation, performed under Mark's leadership, involving the incredibly sophisticated algorithms and ideas of the other co-authors (Stephen Donnelly, Noam Elkies, Tom Fisher,and Nick Rogers). The evidence is as compelling as we have any right to hope for, suggesting that the quadratic twists all have rank less than or equal to 7.

We explain the Bhargava composition algorithm for binary quadratic forms, in historical context

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