On congruent primes and class numbers of imaginary quadratic fields

Tom 159 / 2013

Nils Bruin, Brett Hemenway Acta Arithmetica 159 (2013), 63-87 MSC: Primary 11G05; Secondary 11R29. DOI: 10.4064/aa159-1-4


We consider the problem of determining whether a given prime $p$ is a congruent number. We present an easily computed criterion that allows us to conclude that certain primes for which congruency was previously undecided, are in fact not congruent. As a result, we get additional information on the possible sizes of Tate–Shafarevich groups of the associated elliptic curves.

We also present a related criterion for primes $p$ such that $16$ divides the class number of the imaginary quadratic field $\mathbb {Q}(\sqrt {-p})$. Both results are based on descent methods.

While we cannot show for either criterion individually that there are infinitely many primes that satisfy it nor that there are infinitely many that do not, we do exploit a slight difference between the two to conclude that at least one of the criteria is satisfied by infinitely many primes.


  • Nils BruinDepartment of Mathematics
    Simon Fraser University
    Burnaby, BC V5A 1S6, Canada
  • Brett HemenwayDepartment of Mathematics
    University of Michigan
    Ann Arbor, MI 48109-1043, U.S.A.

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