Reference request: Oldest number theory books with (unsolved) exercises? The 2019 Stack Overflow Developer Survey Results Are InClassical Enumerative Geometry ReferencesHow does “modern” number theory contribute to further understanding of $mathbbN$?Divergent Series as a topic of researchGeometric intuition for Fontaine-Wintenberger?Classification of singularities of plane curves of fixed degree (reference request)Reference request: Oldest calculus, real analysis books with exercises?Reference request: Oldest linear algebra books with exercises?Reference request for bounds of $n$-th compositeReference request: Oldest complex analysis books with (unsolved) exercises?Reference request: Oldest (non-analytic) geometry books with (unsolved) exercises?

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



The 2019 Stack Overflow Developer Survey Results Are InClassical Enumerative Geometry ReferencesHow does “modern” number theory contribute to further understanding of $mathbbN$?Divergent Series as a topic of researchGeometric intuition for Fontaine-Wintenberger?Classification of singularities of plane curves of fixed degree (reference request)Reference request: Oldest calculus, real analysis books with exercises?Reference request: Oldest linear algebra books with exercises?Reference request for bounds of $n$-th compositeReference request: Oldest complex analysis books with (unsolved) exercises?Reference request: Oldest (non-analytic) geometry books with (unsolved) exercises?










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.










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
    9 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
    3 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.










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
    9 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
    3 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.










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






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
    9 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
    3 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
    9 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
    3 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
9 hours ago





$begingroup$
Not sure if there are exercises: books.google.com/books/about/…
$endgroup$
– Cherng-tiao Perng
9 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
3 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
3 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









$endgroup$













    Your Answer





    StackExchange.ifUsing("editor", function ()
    return StackExchange.using("mathjaxEditing", function ()
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    );
    );
    , "mathjax-editing");

    StackExchange.ready(function()
    var channelOptions =
    tags: "".split(" "),
    id: "504"
    ;
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function()
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled)
    StackExchange.using("snippets", function()
    createEditor();
    );

    else
    createEditor();

    );

    function createEditor()
    StackExchange.prepareEditor(
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader:
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    ,
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    );



    );













    draft saved

    draft discarded


















    StackExchange.ready(
    function ()
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f327697%2freference-request-oldest-number-theory-books-with-unsolved-exercises%23new-answer', 'question_page');

    );

    Post as a guest















    Required, but never shown

























    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    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















    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 6 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



























            draft saved

            draft discarded
















































            Thanks for contributing an answer to MathOverflow!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid


            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.

            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f327697%2freference-request-oldest-number-theory-books-with-unsolved-exercises%23new-answer', 'question_page');

            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            Popular posts from this blog

            Are there any AGPL-style licences that require source code modifications to be public? Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?Force derivative works to be publicAre there any GPL like licenses for Apple App Store?Do you violate the GPL if you provide source code that cannot be compiled?GPL - is it distribution to use libraries in an appliance loaned to customers?Distributing App for free which uses GPL'ed codeModifications of server software under GPL, with web/CLI interfaceDoes using an AGPLv3-licensed library prevent me from dual-licensing my own source code?Can I publish only select code under GPLv3 from a private project?Is there published precedent regarding the scope of covered work that uses AGPL software?If MIT licensed code links to GPL licensed code what should be the license of the resulting binary program?If I use a public API endpoint that has its source code licensed under AGPL in my app, do I need to disclose my source?

            2013 GY136 Descoberta | Órbita | Referências Menu de navegação«List Of Centaurs and Scattered-Disk Objects»«List of Known Trans-Neptunian Objects»

            Mortes em março de 2019 Referências Menu de navegação«Zhores Alferov, Nobel de Física bielorrusso, morre aos 88 anos - Ciência»«Fallece Rafael Torija, o bispo emérito de Ciudad Real»«Peter Hurford dies at 88»«Keith Flint, vocalista do The Prodigy, morre aos 49 anos»«Luke Perry, ator de 'Barrados no baile' e 'Riverdale', morre aos 52 anos»«Former Rangers and Scotland captain Eric Caldow dies, aged 84»«Morreu, aos 61 anos, a antiga lenda do wrestling King Kong Bundy»«Fallece el actor y director teatral Abraham Stavans»«In Memoriam Guillaume Faye»«Sidney Sheinberg, a Force Behind Universal and Spielberg, Is Dead at 84»«Carmine Persico, Colombo Crime Family Boss, Is Dead at 85»«Dirigent Michael Gielen gestorben»«Ciclista tricampeã mundial e prata na Rio 2016 é encontrada morta em casa aos 23 anos»«Pagan Community Notes: Raven Grimassi dies, Indianapolis pop-up event cancelled, Circle Sanctuary announces new podcast, and more!»«Hal Blaine, Wrecking Crew Drummer, Dies at 90»«Morre Coutinho, que editou dupla lendária com Pelé no Santos»«Cantor Demétrius, ídolo da Jovem Guarda, morre em SP»«Ex-presidente do Vasco, Eurico Miranda morre no Rio de Janeiro»«Bronze no Mundial de basquete de 1971, Laís Elena morre aos 76 anos»«Diretor de Corridas da F1, Charlie Whiting morre aos 66 anos às vésperas do GP da Austrália»«Morreu o cardeal Danneels, da Bélgica»«Morreu o cartoonista Augusto Cid»«Morreu a atriz Maria Isabel de Lizandra, de "Vale Tudo" e novelas da Tupi»«WS Merwin, prize-winning poet of nature, dies at 91»«Atriz Márcia Real morre em São Paulo aos 88 anos»«Mauritanie: décès de l'ancien président Mohamed Mahmoud ould Louly»«Morreu Dick Dale, o rei da surf guitar e de "Pulp Fiction"»«Falleció Víctor Genes»«João Carlos Marinho, autor de 'O Gênio do Crime', morre em SP»«Legendary Horror Director and SFX Artist John Carl Buechler Dies at 66»«Morre em Salvador a religiosa Makota Valdina»«مرگ بازیکن‌ سابق نساجی بر اثر سقوط سنگ در مازندران»«Domingos Oliveira morre no Rio»«Morre Airton Ravagniani, ex-São Paulo, Fla, Vasco, Grêmio e Sport - Notícias»«Morre o escritor Flavio Moreira da Costa»«Larry Cohen, Writer-Director of 'It's Alive' and 'Hell Up in Harlem,' Dies at 77»«Scott Walker, experimental singer-songwriter, dead at 76»«Joseph Pilato, Day of the Dead Star and Horror Favorite, Dies at 70»«Sheffield United set to pay tribute to legendary goalkeeper Ted Burgin who has died at 91»«Morre Rafael Henzel, sobrevivente de acidente aéreo da Chapecoense»«Morre Valery Bykovsky, um dos primeiros cosmonautas da União Soviética»«Agnès Varda, cineasta da Nouvelle Vague, morre aos 90 anos»«Agnès Varda, cineasta francesa, morre aos 90 anos»«Tania Mallet, James Bond Actress and Helen Mirren's Cousin, Dies at 77»e