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Per the title, what are some of the oldest number theory books out there with (unsolved) exercises? Maybe there are some hidden gems from before the 20th century out there. I am already aware of the books of Dickson and Hardy.

Motivation for this question. Some person came up to me before class and asked, are you the person asking the ridiculous "oldest book with exercises" questions on MO? I said yes, and he asked I could do one on number theory. So here we are.

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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:

  1. 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.

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    $\begingroup$ Carmichael followed this up with his 1915 book, Diophantine Analysis, which also had exercises at the end of each chapter. $\endgroup$ – Gerry Myerson Apr 10 at 22:15
  • $\begingroup$ Is it a well-known problem that many number theory experts have seriously tried but failed to solve? $\endgroup$ – user21820 Apr 11 at 5:55
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    $\begingroup$ @user, it is a well-known problem. Since we seldom publish our failures, it is hard to know how many experts may have seriously tried but failed to settle it. $\endgroup$ – Gerry Myerson Apr 11 at 6:37
  • $\begingroup$ @GerryMyerson: Okay thanks for the information! $\endgroup$ – user21820 Apr 11 at 6:38
  • $\begingroup$ Here, anyway, is one failed attempt to find a counterexample: ams.org/journals/mcom/1994-63-207/S0025-5718-1994-1226815-3 Aaron Schlafly and Stan Wagon, Carmichael's conjecture on the Euler function is valid below $ 10^{10,000,000}$, Math. Comp. 63 (1994), 415-419. See also B39 in Guy, Unsolved Problems In Number Theory. $\endgroup$ – Gerry Myerson Apr 11 at 6:41
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The book "Théorie des nombres, Tome premier" by Edouard Lucas was published 1891. Many of the "Exemples" are actually exercises left to the reader. A scan is freely available in the archive.

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