About This Blog: Discussions of solved and unsolved problems, current research, and other mathematics related topics.

About Me: Mathematics graduate student and Senior Systems Analyst with a passion for mathematics.

Thanks for visiting Guzman’s Mathematics Weblog! I appreciate comments and will gladly address all questions regarding any of the posts. While, it is my hope that everyone can enjoy and benefit from this blog, I must provide a forewarned message. Please note that some of the material covered (those containing proofs) is for undergraduates, advance undergraduates and graduate mathematics students. I try to keep the posts self-contained, but for the sake of brevity, I assume that you have the following prerequisite knowledge:

Language of Mathematics (Rudiments of Logic)
Set Theory
The Integers
Mathematical Induction
Complex Numbers

Mandelbrot Set
Mandelbrot Set


The art of writing is the art of discovering what you believe.
— Gustave Flaubert, French writer (12 December 1821 – 8 May 1880)

Without mathematics there is no art.
— Luca Pacioli, Italian mathematician (1445 – 1517)

It is not in space that I must seek my human dignity, but in the ordering of my thought. It will do me no good to own land. Through space the universe grasps me and swallows me up like a speck; through thought I grasp it.
— Blaise Pascal, French mathematician (19 June 1623 – 19 August 1662)

Paul [Erdős] talks about The Book. The Book has all the theorems of mathematics. Theorems can be proven in a lot of different ways, but in The Book there is only one proof and it is the one that is the clearest proof, the one that gives the most insight, the most aesthetic proof. It’s what he calls The Book proof. And sometimes when there’s a problem and somebody solves it and the proof is not so beautiful, then he’ll say, “Well okay, but let’s look for The Book proof; let’s try to find The Book proof.” And this is the sense of mathematics, that…that The Book is there, the theorems have an existence of their own. And what we’re doing is we’re just trying to uncover. We’re trying to read the pages of The Book. We don’t create mathematics. What we do is we read the pages of The Book. We discover the pages of The Book. So when he goes from university to university, and he talks about problems, and he asks everybody to try to solve these problems, it doesn’t matter who solves the problem. It really doesn’t matter to him, because all of us are in the same venture. We’re all just trying to uncover the pages. And sometimes we succeed. Sometimes we find these beautiful theorems.
— Joel Spencer, American mathematician (20 April 1946 – )

Julia Set
Julia Set