• Emil Grosswald
Part of the Modern Birkhäuser Classics book series (MBC)


As already mentioned, we assume that the reader is familiar with the properties of the natural integers and those of the rational integers, as well as with the elementary operations. Given any two rational integers, one can always add, subtract, or multiply them, and again obtain as result an integer. We already observed in Chapter 2 that in general this is no longer the case with the operation of division; hence, the following definition is nontrivial.


Great Common Divisor Common Multiple Algebraic Integer Small Positive Integer Rational Integer 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. Brun, La série (1/5) 4- (1/7) + (1/11) + (1/13) +... est convergente ou finie. Bull, des Sciences Math. (2) 43 (1919) 100–104, 124–128.MATHGoogle Scholar
  2. 2.
    P. Erdös, On the difference of consecutive primes. Quarterly J. of Math. (Oxford) 6 (1935) 124–128.CrossRefGoogle Scholar
  3. 3.
    P. Erdös, On some applications of Brun’s method. Acta Sci. Math. Szeged 13 (1949) 57–63.MATHGoogle Scholar
  4. 4.
    G. H. Hardy and J. E. Littlewood, Some Problems of Partitio Numerorum III. Acta Math. 44 (1923) 1–70.CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 3rd ed. Oxford: Clarendon Press, 1954.MATHGoogle Scholar
  6. 6.
    I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers, 4th ed. New York: Wiley, 1980.MATHGoogle Scholar
  7. 7.
    K. Prachar, Ueber ein Resultat von Walfisz. Monatshefte für Mathem. 58 (1954) 114–116.CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    K. Prachar, Primzahlverteilung (Die Grundlehren der Math. Wiss. in Einzeldarst., Vol. 91). Berlin: Springer, 1957.Google Scholar
  9. 9.
    D. Shanks, Solved and Unsolved Problems in Number Theory, Washington, D.C.: Spartan Books, 1962.Google Scholar
  10. 10.
    W. Sierpinski, Remarques sur la répartition des nombres premiers. Colloqu. Math. 1 (1948) 193–194.MATHMathSciNetGoogle Scholar
  11. 11.
    A. Walfisz, Stark isolierte Primzahlen. Doklady Akad. Nauk SSSR 90 (1953) 711–713.MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1984

Authors and Affiliations

  • Emil Grosswald
    • 1
  1. 1.Department of MathematicsTemple UniversityPhiladelphiaUSA

Personalised recommendations