Representing numbers as sums of two squares

  • Martin Aigner
  • Günter M. Ziegler


This question is as old as number theory, and its solution is a classic in the field. The “hard” part of the solution is to see that every prime number of the form 4m + 1 is a sum of two squares. G. H. Hardy writes that this two square theorem of Fermat “is ranked, very justly, as one of the finest in arithmetic.” Nevertheless, one of our Book Proofs below is quite recent.


Equivalence Class Prime Factor Prime Number Euclidean Algorithm Prime Number Theorem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    F. W. Clarke, W. N. Everitt, L. L. Littlejohn & S. J. R. Vorster: H. J. S. Smith and the Fermat Two Squares Theorem, Amer. Math. Monthly 106 (1999), 652–665.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    D. R. Heath-Brown: Fermat’s two squares theorem, Invariant (1984), 2–5.Google Scholar
  3. [3]
    I. Niven & H. S. Zuckerman: An Introduction to the Theory of Numbers, Fifth edition, Wiley, New York 1972.MATHGoogle Scholar
  4. [4]
    H. Riesel: Prime Numbers and Computer Methods for Factorization, Second edition, Progress in Mathematics 126, Birkhäuser, Boston MA 1994.MATHCrossRefGoogle Scholar
  5. [5]
    M. Rubinstein & P. Sarnak: Chebyshev’s bias, Experimental Mathematics 3 (1994), 173–197.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    A. Thue: Et par antydninger til en taltheoretisk metode, Kra. Vidensk. Selsk. Forh. 7 (1902), 57–75.Google Scholar
  7. [7]
    S. Wagon: Editor’s corner: The Euclidean algorithm strikes again, Amer. Math. Monthly 97 (1990), 125–129.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    D. Zagier: A one-sentence proof that every prime p≡ 1 (mod 4) is a sum of two squares, Amer. Math. Monthly 97 (1990), 144.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Martin Aigner
    • 1
  • Günter M. Ziegler
    • 2
  1. 1.Institut für Mathematik II (WE2)Freie Universität BerlinBerlinGermany
  2. 2.Institut für Mathematik, MA 6-2Technische Universität BerlinBerlinGermany

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