

The Ramanujan-Nagell equation, February 26, 2019

Noam Elkies,
Tue, February 26, 6pm - 7pm, Science Center 507.

The Ramanujan-Nagell equation. Abstract: In 1913 Ramanujan asked:
2n-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.