Back at the café, I knew I probably wouldn't get a response anytime soon. My immediate goal was to complete my mission. After spending weeks reading books and conducting research, I decided to delve into the field of number theory. This branch of mathematics focuses on the properties and relationships between whole numbers. Number theory has significant applications in cryptography, theoretical computer science, physics, biology, and other scientific fields. Moreover, it harbors numerous unsolved mathematical problems, some of which are famous, such as the Goldbach conjecture, the Collatz conjecture, and the Riemann hypothesis.
In short, it was exactly what I loved. I opened my text editor and began to write: "General Solution to the Syracuse Conjecture."
The Syracuse conjecture is a seemingly simple mathematical problem: regardless of the initial value, can we always obtain a sequence of numbers that eventually becomes 1 by repeating the following two operations
- If n is even, divide it by 2.
- If n is odd, multiply it by 3 and add 1.
For example, if we start with the number 1, the sequence would be as follows: 1, 4, 2, 1.
Indeed, 1 is odd, so we multiply it by 3 and add 1, resulting in 4. 4 is even, so we divide it by 2, yielding 2. 2 is odd, so we multiply it by 3 and add 1, bringing us back to 1.
The Syracuse conjecture asserts that this sequence will always reach 1 regardless of the initial number. However, until now, no one has been able to prove this statement. At least, not until now. I discovered that the deeper I delved into my knowledge of the subject, the less the system demanded mathematical proofs to support my theory.
So, for the following days, I delved even deeper into the mysteries of whole numbers and this seemingly simple yet infinitely complex sequence. I read everything related to my conjecture, and as I read, I began to see connections between the Syracuse conjecture and other mathematical concepts. I was so engrossed in my research that I missed the entire first week of classes, and on the following Monday, I found myself sitting in the office of the head of the mathematics department.
Facing me was Amed, a man in his sixties, a mathematician with an international reputation. In his hands, he held a stack of papers that he read with intense concentration. At first, he had summoned me to discuss my absence, as I had spent my entire day at the library instead of attending my classes. Before he could even start reprimanding me, I apologized and handed him the proof I had worked on all week. Initially, he was skeptical, but the more he read, the more focused he became. After half an hour, he asked :
- So, is this what you've been working on all week?
- Yes, sir.
- You did this all on your own?
- Yes, sir.
- To be honest, you've surprised me. Even if this turns out to be false, the fact that you have new ideas and that it pushed you to understand new concepts is beneficial for you. Let me take a closer look. Take a two-day break and come back to see me afterward. I'll check for errors, and if not, I'll help you publish it.
I left his office feeling somewhat relieved but also exhausted since I had been getting only 3 or 4 hours of sleep per night. I wasn't worried about the validity of the proof. As I exited the office, the system notification rang:
[Mission Completed]
[Condition: Spend the entire day at the library until your first thesis draft is written.]
[Reward: 50 points]