
Noam Elkies,
Tue, February 26, 6pm  7pm, Science Center 507.
The RamanujanNagell equation.
Abstract: In 1913 Ramanujan asked:
2^{n}7 is a perfect square for the values 3, 4, 5, 7, 15 of n.
Find other values.
A few years earlier Thue had proved a result that can be used to show
that there are only finitely many solutions, but  as with several
related finiteness theorems for Diophantine equations  the proof
is "ineffective" in that it cannot be used to guarantee that a list
of solutions is complete. For Ramanujan's problem, it took 35 years
until Nagell proved that there are no other such n. We state several
finiteness theorems, outline some of the connections among them,
explain how a finiteness proof can be ineffective, and (time permitting)
sketch Nagell's proof and an even more elementary one discovered
only 12 years ago by C. Bright.
