By the way, I am also a number theorist.
In my recent answer to another question I described the fickleness of undergraduates in a way which I hope comes across as largely sympathetic. The point of being an undergraduate is to explore the academic terrain a bit, flitting from topic to topic. If you're not a bit flighty you'll have trouble flitting properly, which is more dangerous: you don't want to spend years working on something and discover at the end that you don't like it at all (you want to spend much less time).
However, my answer concerned research, and your project doesn't sound like a research project. It sounds rather like a reading project, or in other words a (largely) independent learning experience. (For those who don't know, "solving polynomials by radicals" is an important topic in mathematics which received a definitive solution in the first half of the 19th century.) For a reading project, you do want to exert some care in choosing something that will be worth the reading.
So I think you should definitely talk to the lecturer right away. He gave you a topic because you didn't have something in mind, now you have something in mind, so it's time to go back. However you should not be surprised to receive an explanation of why solving polynomials by radicals makes a better reading project than the Collatz Conjecture. In the former case, there is an absolutely central question in number theory and algebra, it has a great answer, and all the steps leading up to that answer are also highly central and important. In the latter case there is a rather isolated, curious open problem. It's elementary, innocuous and catchy, which combine to create the psychological impression that it can't be that hard...but it is that hard. In particular I am struck by the statement
I know that as a third year student I definitely do not possess the capabilities of making any sort of impact on the conjecture[.]
To that I reply:
I know that as someone with a PhD from Harvard who has successfully solved some open problems in number theory that I probably do not possess the capabilities of making any sort of impact on the conjecture.
The point being that nobody does, so far as I know. If you think you want to tilt at windmills, it's a good time to check in again on what the assignment was. (This is not to say that you should stop thinking about the Collatz Problem altogether. Rather, if you like it, think about it in your spare time.)
I will end with a story. I taught an advanced undergraduate number theory course twice: see here. Each time the course included a final project. I gave a lot of suggestions for these final projects, and so far as I can remember most students did follow up on one of my suggestions. Certainly that is true with the two best projects, which both led to other (novel!) work by those students later on. The first time I taught the course, a student told me that he wanted to write about what ancient Greeks knew and believed about perfect numbers. I told him honestly that I knew almost nothing about that and therefore couldn't be confident that there was much to say. He replied enthusiastically that there were many things: did I know for instance that the Greeks thought that there were no odd perfect numbers, that there were infinitely many even perfect numbers, and that the final digits in this infinite sequence always alternated between 6 and 8? Well, no, so I let him write on that topic. When I read his paper, I found...well, I have already told you everything I remember on the subject, but it turned out that I had made the wrong decision. Some topics are more fruitful than others.
