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.
Primes in Intervals of Bounded Length
Basics of binary quadratic forms and Gauss composition
Don't be seduced by the zeros!
Different approaches to the distribution of primes
The Princeton Companion to Mathematics: Analytic number theory
Prime number patterns (2009 Ford Prize)
It is easy to determine whether a given integer is prime (2008 Chauvenet Prize)
Prime number races (2007 Ford Prize)
It's as easy as abc
Zaphod Beeblebrox's brain and the fifty-ninth row of Pascal's triangle (1995 Hasse Prize)