I wonder if you are already aware of R. D. Carmichael's "The theory of numbers" (John Wiley & Sons, Inc., NY, pp. 94, 1914.).

Apropos of the exercises in this monograph, one can read the following in the preface:

Numerous problems are supplied throughout the text. These have been
selected with great care so as to serve as excellent exercises for the
student's introductory training in the methods of number theory and to
afford at the same time a further collection of useful results. The
exercises with a star are more difficult than the others; they will
doubtless appeal to the best students.

Among the numerous problems supplied, the eighth problem on page 36 does stand out because, as far as I know, **nobody has been able to solve it yet**. It goes as follows:

- Show that if the equation $$\phi(x) = n$$ has one solution; it always has a second solution, $n$ being given and $x$ being the
unknown.

Oddly enough, Carmichael didn't consider that this question deserved a star... In case you want to learn more about the history of this problem, I recommend that you take a look at the following installment of *The evidence* (a column that Stan Wagon used to contribute to *The Mathematical Intelligencer*):

S. Wagon, Carmichael's "empirical theorem". *Math. Intelligencer*, **8** (1986), No. 2, pp. 61-63.

Elementary Number Theoryby Uspensky and Heaslet has a bunch of problems in each chapter. Not sure whether it's the oldness you are looking for (my impression is that the exercises don't get better if you go older than this). $\endgroup$ – darij grinberg Apr 10 at 17:55