**Note:** Many problems in this text are exploratory in nature.
Nonetheless, you should try to write rigorous proofs whenever possible. Sometimes
this will not be possible - either because you don't see the proof or because
it really is beyond the course. In any case, do the best that you can. On this
particular assignment, you should be able to be very rigorous on exercises 1.2
and 1.3. Exercise 1.1 is more exploratory in nature, so do your best at writing
up your conjecture(s) in a clear way and presenting proofs. Hopefully, over
the course of the semester, you will continue to make improvements in the gathering
data/making conjectures/writing proofs process.

**Another comment: **Problems in number theory can often be very
difficult if you don't happen to see the solution right away. Do not assume
that the exercises will increase monotonically in difficulty as you proceed
through the assignment. If you get stuck on the first problem, then do not get
frustrated and quit. Move on to the second problem or the third problem. Do
what you can and then return to the first problem. Often times, on a second
or third attempt, you will be able to see a solution that first eluded you.
And sometimes work on later problems can provide further insights for work on
earlier problems. Do not work linearly through an assignment. You are not a
computer!

Finally, have fun! These problems really are a lot of fun.