Skip to main content
Mathematics LibreTexts


  • Page ID
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    • 1.1: Compositions and Partitions
    • 1.2: Arithmetic Functions
    • 1.3: Distribution of Primes
      Perhaps the best known proof in all of “real” mathematics is Euclid’s proof of the existence of infinitely many primes.
    • 1.4: Irrational Numbers
    • 1.5: Congruences
      In this section we shall develop some aspects of the theory of divisibility and congruences.
    • 1.6: Diophantine Equations
      Volume 2 of Dickson’s History of the Theory of Numbers deals with Diophantine equations. It is as large as the other two volumes combined. It is therefore clear that we shall not cover much of this ground in this section. We shall confine our attention to some problems which are interesting though not of central importance.
    • 1.7: Combinatorial Number Theory
      There are many interesting questions that lie between number theory and combinatorial analysis. We consider first one that goes back to I. Schur (1917) and is related in a surprising way to Fermat’s Last Theorem.
    • 1.8: Geometry of Numbers
      We have already seen that geometrical concepts are sometimes useful in illuminating number theoretic considerations. With the introduction by Minkowski of geometry of numbers a real welding of important parts of number theory and geometry was achieved. This branch of mathematics has been in considerable vogue in the last 20 years, particularly in England where it was and is being developed vigorously by Mordell, Davenport, Mahler and their students.

    Chapters is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

    • Was this article helpful?