Who is the "father of number theory"?

You can see :

• André Weil, Number Theory : An approach through history (1984), Preface, page ix :

Fermat, Euler, Lagrange, Legendre. These are the founders of modern number theory. The greatness of Gauss lies in his having brought to completion what his predecessors had initiated, no less than in his inaugurating a new era in the history of the subject.

Fermat contributed a lot of results (not so many proofs, unfortunately) to the community, and did lots of work in number theory. For example, both Fermat's Little Theorem and Fermat's Last Theorem are named after him, and they are clearly number theoretic. Fermat is particularly notable in that he worked mostly in isolation (if I remember correctly; if anybody knows better, feel free to correct me), and number theory was not such a prominent field at the time; geometry was more highly looked upon, and was the "in-thing", so-to-speak (for the record, Fermat also did geometric work, including contributing to the foundations of calculus).

Gauss contributed a lot to number theory too, as demonstrated in his book "Disquisitiones Arithmeticae", which I think is still in print. Basically everything about this book is important: first, Gauss' work was excellent, both clarifying old ideas and introducing some new ones. Additionally, the book was essentially the first modern number theory textbook, and I've heard it said before that its existence added a lot of interest to the field.

I'm not a mathematical historian, so that's all I really know. With regards to your presentation, I think a good place to start would be to look more closely at the impact of Disquisitiones Arithmeticae on the mathematical community's view of number theory, and then include both Fermat and Gauss in your presentation, perhaps even offering your own viewpoint on who should be called its "father", if anyone. It's worth noting that the "answer" to the question depends on what you mean by "modern".

Incidentally, it's also worth noting that Diophantine equations are still studied today, and, as the name suggests, the themes there go back to Diophantus, who lived long before Fermat and Gauss; a lot of their contributions are about Diophantine equations (see Fermat's Last Theorem!). Another area of interest related to Diophantus is Diophantine approximation.

Not only this, but sieve theory has come along way from the Sieve of Eratosthenes, which, as far as I know, was one of the first prime sieves talked about, the invention of which was attributed to Eratosthenes, another ancient Greek, by Nicomachus.

While these last two contributions clearly aren't really "modern", it's worth noting that the core ideas of modern number theory aren't necessarily all that young; and if you want to take more than the core ideas into account, then you might well consider taking some ideas post-Gauss into account too.