Reference request: Oldest number theory books with (unsolved) exercises? The 2019 Stack...



Reference request: Oldest number theory books with (unsolved) exercises?



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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.






nt.number-theory reference-request







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  • 2




    $begingroup$
    Elementary Number Theory by 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
    9 hours ago






  • 1




    $begingroup$
    Not sure if there are exercises: books.google.com/books/about/…
    $endgroup$
    – Cherng-tiao Perng
    8 hours ago








  • 1




    $begingroup$
    I found this: "Introduzione alla teoria dei numeri, con numerosi esercizi e con notizie storiche." by Vittorio Murer, from 1909 zbmath.org/?q=an%3A40.0266.01
    $endgroup$
    – EFinat-S
    2 hours ago


















3












$begingroup$


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.






nt.number-theory reference-request







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    Elementary Number Theory by 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
    9 hours ago






  • 1




    $begingroup$
    Not sure if there are exercises: books.google.com/books/about/…
    $endgroup$
    – Cherng-tiao Perng
    8 hours ago








  • 1




    $begingroup$
    I found this: "Introduzione alla teoria dei numeri, con numerosi esercizi e con notizie storiche." by Vittorio Murer, from 1909 zbmath.org/?q=an%3A40.0266.01
    $endgroup$
    – EFinat-S
    2 hours ago
















3












3








3





$begingroup$


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.






nt.number-theory reference-request







share|cite|improve this question











$endgroup$




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.






nt.number-theory reference-request




nt.number-theory reference-request






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 9 hours ago







Get Off The Internet

















asked 9 hours ago









Get Off The InternetGet Off The Internet

364320




364320








  • 2




    $begingroup$
    Elementary Number Theory by 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
    9 hours ago






  • 1




    $begingroup$
    Not sure if there are exercises: books.google.com/books/about/…
    $endgroup$
    – Cherng-tiao Perng
    8 hours ago








  • 1




    $begingroup$
    I found this: "Introduzione alla teoria dei numeri, con numerosi esercizi e con notizie storiche." by Vittorio Murer, from 1909 zbmath.org/?q=an%3A40.0266.01
    $endgroup$
    – EFinat-S
    2 hours ago
















  • 2




    $begingroup$
    Elementary Number Theory by 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
    9 hours ago






  • 1




    $begingroup$
    Not sure if there are exercises: books.google.com/books/about/…
    $endgroup$
    – Cherng-tiao Perng
    8 hours ago








  • 1




    $begingroup$
    I found this: "Introduzione alla teoria dei numeri, con numerosi esercizi e con notizie storiche." by Vittorio Murer, from 1909 zbmath.org/?q=an%3A40.0266.01
    $endgroup$
    – EFinat-S
    2 hours ago










2




2




$begingroup$
Elementary Number Theory by 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
9 hours ago




$begingroup$
Elementary Number Theory by 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
9 hours ago




1




1




$begingroup$
Not sure if there are exercises: books.google.com/books/about/…
$endgroup$
– Cherng-tiao Perng
8 hours ago






$begingroup$
Not sure if there are exercises: books.google.com/books/about/…
$endgroup$
– Cherng-tiao Perng
8 hours ago






1




1




$begingroup$
I found this: "Introduzione alla teoria dei numeri, con numerosi esercizi e con notizie storiche." by Vittorio Murer, from 1909 zbmath.org/?q=an%3A40.0266.01
$endgroup$
– EFinat-S
2 hours ago






$begingroup$
I found this: "Introduzione alla teoria dei numeri, con numerosi esercizi e con notizie storiche." by Vittorio Murer, from 1909 zbmath.org/?q=an%3A40.0266.01
$endgroup$
– EFinat-S
2 hours ago












2 Answers
2






active

oldest

votes


















6












$begingroup$

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.






share|cite|improve this answer











$endgroup$









  • 1




    $begingroup$
    Carmichael followed this up with his 1915 book, Diophantine Analysis, which also had exercises at the end of each chapter.
    $endgroup$
    – Gerry Myerson
    5 hours ago



















2












$begingroup$

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.






share|cite|improve this answer









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    2 Answers
    2






    active

    oldest

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    2 Answers
    2






    active

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    active

    oldest

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    6












    $begingroup$

    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.






    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      Carmichael followed this up with his 1915 book, Diophantine Analysis, which also had exercises at the end of each chapter.
      $endgroup$
      – Gerry Myerson
      5 hours ago
















    6












    $begingroup$

    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.






    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      Carmichael followed this up with his 1915 book, Diophantine Analysis, which also had exercises at the end of each chapter.
      $endgroup$
      – Gerry Myerson
      5 hours ago














    6












    6








    6





    $begingroup$

    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.






    share|cite|improve this answer











    $endgroup$



    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.







    share|cite|improve this answer














    share|cite|improve this answer



    share|cite|improve this answer








    edited 3 hours ago

























    answered 5 hours ago









    José Hdz. Stgo.José Hdz. Stgo.

    5,31734877




    5,31734877








    • 1




      $begingroup$
      Carmichael followed this up with his 1915 book, Diophantine Analysis, which also had exercises at the end of each chapter.
      $endgroup$
      – Gerry Myerson
      5 hours ago














    • 1




      $begingroup$
      Carmichael followed this up with his 1915 book, Diophantine Analysis, which also had exercises at the end of each chapter.
      $endgroup$
      – Gerry Myerson
      5 hours ago








    1




    1




    $begingroup$
    Carmichael followed this up with his 1915 book, Diophantine Analysis, which also had exercises at the end of each chapter.
    $endgroup$
    – Gerry Myerson
    5 hours ago




    $begingroup$
    Carmichael followed this up with his 1915 book, Diophantine Analysis, which also had exercises at the end of each chapter.
    $endgroup$
    – Gerry Myerson
    5 hours ago











    2












    $begingroup$

    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.






    share|cite|improve this answer









    $endgroup$


















      2












      $begingroup$

      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.






      share|cite|improve this answer









      $endgroup$
















        2












        2








        2





        $begingroup$

        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.






        share|cite|improve this answer









        $endgroup$



        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.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 2 hours ago









        EFinat-SEFinat-S

        1,2041417




        1,2041417






























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