(
hour) - Student presentations: proof of FLT for exponent 4 and
no square triangles.
(10 minutes) - break
(30 minutes) - statement of Tunnell's criterion.
Let
Let
and
Theorem 7.1 (Waldspurger, Tunnell)
Suppose
is a squarefree integer. If
is odd
then
if and only if the
th coefficient
of
is 0
. If
is even then
if and only if the
th coefficient of
is 0
.
This is a very deep theorem. It allows us to determine
whether or not
.
The Birch and Swinnerton-Dyer conjecture (which is not
a theorem) then asserts that
if and only if
is a congruent number. Thus once enough of the
Birch and Swinnerton-Dyer conjecture is proved, we'll have
an elementary way to decide whether or not a (squarefree)
integer
is a congruent number. Namely,
if
is odd then (conjecturally)
is a congruent number
if and only if
Simiarly,
if
is even then (conjecturally)
is a congruent number
if and only if
(30 minutes) - exercises with Tunnell's criterion.
Come up with a method to explicitly list
each of the above sets.
Prove that the elementary criterion (involving cardinality of
sets) implies that none of
are congruent numbers.
Prove that the elementary criterion (involving cardinality of
sets) implies that
are congruent numbers. Then find
a right triangle with area
.
Verify with SAGE that Theorem 7.1 (appears to)
hold for
. Hint: The command
f1, f2 = tunnell_forms(30)
computes the
and
defined above, and e.g., f1[3]
returns the coefficient of
.
Question (Nathan Ryan): Is it possible to decide whether
or not a prime number
is a (conjectural) congruent number in
time polynomial in the number of digits of
? I.e., if
has a
hundred digits is there any hope we could tell whether or not
is
(conjecturally) a congruent number i a reasonable amount of time?
[[WARNING: This is an unsolved problem, as far as I
know.]]