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



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 $mathbb{N}$?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.






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












$begingroup$


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









$endgroup$














    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

            What is the “three and three hundred thousand syndrome”?Who wrote the book Arena?What five creatures were...

            Image
            What happens when the centripetal force is equal and opposite to the centrifugal force? Why is "la Gestapo" feminine? Nested Dynamic SOQL Query How are passwords stolen from companies if they only store hashes? Exposing a company lying about themselves in a tightly knit industry: Is my career at risk on the long run? Can "few" be used as a subject? If so, what is the rule? Why are there no stars visible in cislunar space? categorizing a variable turns it from insignificant to significant Do I need an EFI partition for each 18.04 ubuntu I have on my HD? What are the consequences of changing the number of hours in ...
            Read more

            Gersau Kjelder | Navigasjonsmeny46°59′0″N 8°31′0″E46°59′0″N...

            Image
            Tidlegare republikkarSkipingar i 1433Kommunar i SchwyzDistrikt i SchwyzNedleggingar i 1798Stader ved Vierwaldstättersee kommunekantonenSchwyzSveitskm²VierwaldstätterseeLauerz Gersau Kommunevåpenet til Gersau Plassering Styresmakter Land   Sveits Kanton kantonen Schwyz Distrikt Gersau Språk tysk Geografi Flatevidd  - kommune 23,70 km² Innbyggjarar  - kommune (desember 2010)    - folketettleik 2 094   88,4 /km² Koordinatar 46°59′0″N 8°31′0″E  Koordinatar: 46°59′0″N 8°31′0″E  Høgd over havet 435 moh Diverse annan informasjon Postnummer 6442 Nettstad: www.gersau.ch Freistaat und Republik Gersau Fristaten og republikken Gersau By-republikk og alliert protektorat i Det gamle sveitsiske eidsforundet ←   ← 1433–1798 1814–1818 →   → Hovudstad Gersau Styreform Republikk Historisk periode Tidleg moderne tid  - Kjøpte fridom frå ...
            Read more

            Hestehale Innhaldsliste Hestehale på kvinner | Hestehale på menn | Galleri | Sjå òg |...

            Image
            Frisyrar frisyrehårhårstrikkhårklemmerøyrethesthovudverkeuropeisk motejobbtreningfletterparykkarungdomsopprørmotkulturmusikararmotorsykkelentusiastarpolitietpolitiskforretningverksemdfiskeflette Hestehale sett bakfrå. Hestehale på ein mannleg indianarleiar. Hestehale er ein frisyre der ein samlar det meste eller alt av langt hår [1] i bakhovudet med ein hårstrikk eller liknande. Avhengig av mote kan det òg vere vanleg å sette opp hestehalen med hårklemmer eller å ha hestehalen hengande på den eine sida av hovudet ved å samle håret ved øyret. Hestehale er ein vanleg frisyre for jenter og for gutar med langt hår. Han har fått namnet sitt etter likskapen med halen på ein hest. For stramme hestehaler kan føre til hårtap, [2] i nokre høve også hovudverk. [3] Innhaldsliste 1 Hestehale på kvinner 2 Hestehale på menn 3 Galleri 4 Sjå òg 5 Bakgrunnsstoff 6 Kjelder Hestehale på kvinner | Ungjente med hestehale som held håret unna ansiktet...
            Read more