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Preface To the Student

Hi! Not too many students read this bit in textbooks, but I hope you do, and I hope you circle stuff you think is important. In pen.
Doing math without writing in the book (or on something, if you’re only using an electronic version) is sort of like reading much literature (like Shakespeare or Homer) or many religious texts (like the Psalms or Vedas) without paying attention to the spoken aspect. It’s possible, and we all may have done it (some successfully), but it’s sort of missing the point.
So read this book and write in it. My students do. They even like it.
Here are three things that will lead to success with this book.
  • You should like exploring numbers and playing with them. If you were the kind of kid who added
    \begin{equation*} 1+2+3+4+5+6+7+8+9+10+\cdots \end{equation*}
    on your calculator when you were bored to see if there would be an interesting pattern, and actually liked it, you will like number theory. If you then tried
    \begin{equation*} 2\cdot 3\cdot 4\cdot 5\cdot 6\cdot 7\cdot 8\cdot 9\cdot 10\cdot \cdots \end{equation*}
    you will really like it.
  • I also hope you are open to using computers to explore math and check conjectures. As Picasso said, “They can only give you answers” – but oh what answers! We use the SageMath 9  system, one that will grow with you and that will always be free to use (for several meanings 10  of the word free 11 ). You don’t have to know how to program to use this, though it’s useful. Plus, you are using number theory under the hood anyway if you use the internet much, so why not?
  • Finally, you should want to know why things are true. I assume a standard introduction to proof course as background, but different people are ready in different ways for this. If you are reasonably familiar with proofs by induction and contradiction, and have some basic experience with sets and relations, that is a good start. Some good free resources online include A Gentle Introduction to the Art of Mathematics [E.3.2] and Book of Proof [E.3.1].
    Some of the proofs will be hairy, and some exercises challenging. (Not all!) Do not worry; by trying, you will get better at explaining why things are true that you are convinced of. And that is a very useful skill. (Provided you are convinced of them; if not, go back to the first bullet point and play with more examples!)

Remark 0.0.1. A final note to the student.

As a last note before you dig in, if you think that it is worth exploring the possible truth (see Section 25.3) of
\begin{equation*} 1+2+3+4+5+6+7+8+9+10+\cdots=-\frac{1}{12}\text{,} \end{equation*}
or if as a kid you did
\begin{equation*} 2^{3^{4^{5^{6^{7^{8^{9^{\cdots}}}}}}}} \end{equation*}
to see what would happen, then maybe you should become a mathematician. In that case, click on all links in the text and find a cool problem that interests you!