The Collatz Conjecture is a very simply stated problem. It’s an still an open problem that simultaneously entertaining and vexing. Terry Tao has a proof for “almost all numbers”. The following video does a fine job of explaining it. The Read More …
A congruence for the Ramanujan Tau function.
Here is a fun problem from Apostol’s “Modular Functions and Dirichlet Series in Number Theory” that involves one of Ramanujan’s conjectures. : Let Prove that First we note is a cusp form of weight 12. Then we write out the Read More …
Torsion Subgroups
Let be an abelian group. Prove that is a subgroup of (called the torsion subgroup of ). Give an explicit example where this set is not a subgroup when is non-abelian. A solution: Let . Since the identity is Read More …
An interesting group
I like algebra. MAT5420 was my first mathematics course in Wayne State. Here’s a problem that gave me entertainment this semester. It’s not from MAT5420. It’s an exercise from Serre’s “Trees”. Show that the group is trivial. Here is one way Read More …