#
LMS Durham Symposium on Computational Number Theory

24 July - 3 August 2000

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Fermat Quartics and a Challenge Curve of Serre

Victor FLYNN (Liverpool)

**Abstract:** A study of rational points on Fermat quartics of
the form *X*^4 + *Y*^4 = *c* immediately reveals many
values of *c* which can be dismissed by congruence
considerations. Many other values of *c* can be dismissed if one
of two associated elliptic curves has rank 0. What remains are the
stubborn values of *c* which cannot be trivially dismissed:
*c* = 17, 82, 97 and 257 being the only such less than 300. A
solution of the case *c* = 17 (posed as a challenge by Serre) is
presented, representing the first success with a nontrivial value of
*c*, and we discuss the extent to which the method might hope to
solve other difficult values of *c*. The talk is based on joint
work with Joe Wetherell.

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*Page maintained by John Cremona: *
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John.Cremona@nottingham.ac.uk
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