This post is going to be strange! It is a nostalgia for me to think about prime numbers and prime factorization algorithms. Maybe prime numbers was the first scientific challenge of my school life, which I faced at high school. I couldn't accept the fact that there is no formula or function to find *nth* prime number like
Prime(n). Later I understood that most of cryptographic algorithms are based on the fact that there is no fast algorithm that is capable of factorizing large numbers consisting of just 2 big prime numbers.

I remember many days at that time, putting paper in front of me and listing lots of prime numbers to find the relation between them. After a while I found a relation between prime numbers and * Repunit *numbers which finally satisfied me to let the math go! and start computer science instead!! I claimed a theorem about it and also proved it.

So what are Repunit numbers?

So the theorem which I proved was this:

After 4 years, when I was graduating my B.Sc., Ayaz (my beloved master) encouraged me to write an article about it, and I did it. I was so excited to wrap it up in a paper, but unfortunately it never got out. But now, after about 10 years, I decided to publish it here in this blog.

Again, I want to thank Ayaz who made me to think about this problem and pushed me forward to write this paper.

You can download the full article from here.