Reference request: Oldest number theory 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.






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
    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
    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
    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
    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
    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
    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
    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
    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
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
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









$endgroup$














    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 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






























            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

            Masuk log Menu navigasi

            Image
            JawaSundaBanyumasanBugisAcehBanjarMinangMelayuDeutschEnglishEspañolFrançaisItalianoNederlands (function()var node=document.getElementById("mw-dismissablenotice-anonplace");if(node)node.outerHTML="u003Cdiv class="mw-dismissable-notice"u003Eu003Cdiv class="mw-dismissable-notice-close"u003E[u003Ca tabindex="0" role="button"u003Etutupu003C/au003E]u003C/divu003Eu003Cdiv class="mw-dismissable-notice-body"u003Eu003Cdiv id="localNotice" lang="id" dir="ltr"u003Eu003C/divu003Eu003C/divu003Eu003C/divu003E";()); Masuk log Loncat ke navigasi Loncat ke pencarian Bahasa: Jawa | Sunda | Banyumasan | Bugis | Aceh | Banjar | Minang | Melayu | Deutsch | English | Español | Français | Italiano | Nederlands Nama pengguna Kata sandi   Biarkan saya tetap masuk Masuk log Bantuan masuk log Lupa kata sandi? Belum punya akun?Gabung Wikipedia Diperoleh dari "https:/...
            Read more

            Identifying “long and narrow” polygons in with PostGISlength and width of polygonWhy postgis st_overlaps reports Qgis' “avoid intersections” generated polygon as overlapping with others?Adjusting polygons to boundary and filling holesDrawing polygons with fixed area?How to remove spikes in Polygons with PostGISDeleting sliver polygons after difference operation in QGIS?Snapping boundaries in PostGISSplit polygon into parts adding attributes based on underlying polygon in QGISSplitting overlap between polygons and assign to nearest polygon using PostGIS?Expanding polygons and clipping at midpoint?Removing Intersection of Buffers in Same Layers

            Image
            Do I have to worry about players making “bad” choices on level up? Is thermodynamics only applicable to systems in equilibrium? gnu parallel how to use with ffmpeg Has any spacecraft ever had the ability to directly communicate with civilian air traffic control? Reverse the word in a string with the same order in javascript Is creating your own "experiment" considered cheating during a physics exam? How to stop co-workers from teasing me because I know Russian? What is a Recurrent Neural Network? Any examples of headwear for races with animal ears? Confusion about capacitors TikZ how to make supply and demand arrows for nodes? Does a creature that is immune to a condition still make a saving throw? How to verbalise code in Mathematica? Upright [...] in italics quotation Cannot populate data in lightning data table Stark VS Thanos Binary Numbers Magic Trick In gnome-terminal only 2 out of 3 zoom keys work You look catfish vs You look like a...
            Read more

            Старые Смолеговицы Содержание История | География | Демография | Достопримечательности | Примечания | НавигацияHGЯOLHGЯOL41 206 832 01641 606 406 141Административно-территориальное деление Ленинградской области«Переписная оброчная книга Водской пятины 1500 года», С. 793«Карта Ингерманландии: Ивангорода, Яма, Копорья, Нотеборга», по материалам 1676 г.«Генеральная карта провинции Ингерманландии» Э. Белинга и А. Андерсина, 1704 г., составлена по материалам 1678 г.«Географический чертёж над Ижорскою землей со своими городами» Адриана Шонбека 1705 г.Новая и достоверная всей Ингерманландии ланткарта. Грав. А. Ростовцев. СПб., 1727 г.Топографическая карта Санкт-Петербургской губернии. 5-и верстка. Шуберт. 1834 г.Описание Санкт-Петербургской губернии по уездам и станамСпецкарта западной части России Ф. Ф. Шуберта. 1844 г.Алфавитный список селений по уездам и станам С.-Петербургской губернииСписки населённых мест Российской Империи, составленные и издаваемые центральным статистическим комитетом министерства внутренних дел. XXXVII. Санкт-Петербургская губерния. По состоянию на 1862 год. СПб. 1864. С. 203Материалы по статистике народного хозяйства в С.-Петербургской губернии. Вып. IX. Частновладельческое хозяйство в Ямбургском уезде. СПб, 1888, С. 146, С. 2, 7, 54Положение о гербе муниципального образования Курское сельское поселениеСправочник истории административно-территориального деления Ленинградской области.Топографическая карта Ленинградской области, квадрат О-35-23-В (Хотыницы), 1930 г.АрхивированоАдминистративно-территориальное деление Ленинградской области. — Л., 1933, С. 27, 198АрхивированоАдминистративно-экономический справочник по Ленинградской области. — Л., 1936, с. 219АрхивированоАдминистративно-территориальное деление Ленинградской области. — Л., 1966, с. 175АрхивированоАдминистративно-территориальное деление Ленинградской области. — Лениздат, 1973, С. 180АрхивированоАдминистративно-территориальное деление Ленинградской области. — Лениздат, 1990, ISBN 5-289-00612-5, С. 38АрхивированоАдминистративно-территориальное деление Ленинградской области. — СПб., 2007, с. 60АрхивированоКоряков Юрий База данных «Этно-языковой состав населённых пунктов России». Ленинградская область.Административно-территориальное деление Ленинградской области. — СПб, 1997, ISBN 5-86153-055-6, С. 41АрхивированоКультовый комплекс Старые Смолеговицы // Электронная энциклопедия ЭрмитажаПроблемы выявления, изучения и сохранения культовых комплексов с каменными крестами: по материалам работ 2016-2017 гг. в Ленинградской области

            Image
            Населённые пункты по алфавитуНаселённые пункты Волосовского районаМызы Ингрии Беседском сельском поселенииВолосовского районаЛенинградской областиПисцово...
            Read more