Groups and Homomorphisms

One often sees that proving theorems about groups and related structures can involve clever reasoning and ingenuity. I will be collecting, when possible, problems which use such reasoning, because what is more disorienting than not being able to re-use a clever method you discovered a few weeks ago? Problem 1: Show that has the same number…

Results about Finite-Dimensional Linear Spaces

Just like my page for recording the results I encounter about multiplicative functions, along with their proofs, I intend to update this page with basic results about linear spaces (mostly finite-dimensional). I will try to explicitly state the linear algebra theorems I make use of in my proofs. Result 1: If is a finite-dimensional vector space,…

Results about Multiplicative functions

I will be regularly updating this post with results, along with my proofs of them, that I encounter about multiplicative functions, in particular involving the Dirichlet product. A multiplicative function is such that for , , where . The Dirichlet product of two arithmetical functions (functions that take operate in the integer domain)  is . Result 1:…

A Construction Comes in Handy

One of my friends shared a geometry problem with me, already quite despondent after having tried it for long. The problem statement was : In the square below,  . Prove that . Unsurprisingly, I spent the first twenty minutes applying trigonometry and classical geometry to solve the problem. However, when the size of the calculations…

Number Theory : The Queen of Mathematics

The Prince of Mathematics, Carl Friedrich Gauss once famously said : Mathematics is the Queen of the Sciences and Number theory is the Queen of Mathematics. This statement though highly debatable, does find resonance in many of the great mathematicians of both the past and the present (and most probably this will be true of…