5 Conjectures That Motivated Me To Do A PhD In Number Theory.

This article is meant not as a deep dive into each of these conjectures, but more so as a taster for some incredibly fascinating problems. As we will see, some of these problems are astonishingly simple to understand, yet their proof eludes even the brightest of minds. It is hard to imagine that anyone who has even the slightest interest in mathematics would not feel a burning urge to give some of these a try. I hope that, just like me, you find each of these a fascinating example of how even the simplest problems can be incredibly difficult to solve.

The Collatz Conjecture, also known as the 3n+1 conjecture was first proposed in 1937 by German mathematician Lothar Collatz. The conjecture states that if you start with any positive integer, and repeatedly apply the following two rules:

then you will eventually reach the number 1.

For example, if you start with the number 17, you would follow these steps:

17 → 52 → 26 → 13 → 40 → 20 → 10 → 5 → 16 → 8 → 4 → 2 → 1.

As you can see, you eventually reach the number 1.

The Collatz conjecture has been tested for numbers up to 2⁶⁸, and it has always been found to be true. Similarly, it has been meticulously examined and tested for vast collections of numbers, with no counterexamples discovered. It is hard to deny that it is the elusive nature of a solution to this conjecture which leaves mathematicians enthralled.

To date, the most breakthrough work has been carried out by Terence Tao who in 2019 showed that the conjecture is true for almost all numbers.

Note: The terminology “almost all numbers” has a very specific meaning in mathematics. If you have not come across the terminology before, it can be a little complicated at first to understand and appreciate it, but, in a nutshell, it basically means that the conjecture holds true for all positive integers except for a very small set. Interestingly, the exact size of this small set of numbers is very difficult to determine, but importantly it is not zero…

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