Advertisement

Fermat: The Founder of Modern Number Theory

  • Israel Kleiner
Chapter

Abstract

Fermat, though a lawyer by profession and only an “amateur” mathematician, is regarded as the founder of modern number theory. What were some of his major results in that field? What inspired his labors? Why did he not publish his proofs? How did scholars attempt to reconstruct them? Did Fermat have a proof of Fermat’s Last Theorem? What were the attitudes of seventeenth-century mathematicians to his number theory? These are among the questions we will address in this chapter.

Keywords

Number Theory Elliptic Curve Elliptic Curf Diophantine Equation Binary Quadratic Form 
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.

References

  1. 1.
    W. W. Adams and L. J. Goldstein, Introduction to Number Theory, Prentice-Hall, 1976.Google Scholar
  2. 2.
    A. G. Agargün and E. M. Özkan, A historical survey of the Fundamental Theorem of Arithmetic, Hist. Math. 28 (2001) 207-214.CrossRefMATHGoogle Scholar
  3. 3.
    E. Bach and J. Shallit, Algorithmic Number Theory, Vol. 1, MIT Press, 1996.Google Scholar
  4. 4.
    A. Baker, Transcendental Number Theory, Cambridge Univ. Press, 1990.MATHGoogle Scholar
  5. 5.
    K. Barner, How old did Fermat become?, NTM, Intern. Jour. Hist. and Ethics of Natur. Sc., Techn. and Med. 8 (4) (October 2001).Google Scholar
  6. 6.
    E. J. Barbeau, Pell’s Equation, Springer, 2003.Google Scholar
  7. 7.
    I. G. Bashmakova, Diophantus and Diophantine Equations, Math. Assoc. of Amer., 1997. (Translated from the Russian by A. Shenitzer.)Google Scholar
  8. 8.
    E. T. Bell, Men of Mathematics, Simon and Schuster, 1937.Google Scholar
  9. 9.
    F. Bornemann, PRIMES is in P: A breakthrough for ‘everyman’, Notices of the Amer. Math. Soc. 50 (2003) 545-552.MATHMathSciNetGoogle Scholar
  10. 10.
    D. M. Bressoud, Factorization and Primality Testing, Springer, 1989.Google Scholar
  11. 11.
    D. A. Cox, Primes of the Formx 2 + ny 2: Fermat, Class Field Theory, and Complex Multiplication, Wiley, 1989.Google Scholar
  12. 12.
    L. E. Dickson, History of the Theory of Numbers, 3 vols., Chelsea, 1966.Google Scholar
  13. 13.
    H. M. Edwards, Fermat’s Last Theorem: A Genetic Introduction to Algebraic Number Theory, Springer, 1977.Google Scholar
  14. 14.
    C. R. Fletcher, A reconstruction of the Frenicle-Fermat correspondence, Hist. Math. 18 (1991) 344–351.CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    C. R. Fletcher, Fermat’s theorem, Hist. Math. 16 (1989) 149–153.CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    K. Fogarty and C. O’Sullivan, Arithmetic progressions with three parts in prescribed ratio and a challenge of Fermat, Math. Mag. 77 (2004) 283–292.CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    J. R. Goldman, The Queen of Mathematics: A Historically Motivated Guide to Number Theory, A K Peters, 1998.Google Scholar
  18. 18.
    E. Grosswald, Representation of Integers as Sums of Squares, Springer, 1985.Google Scholar
  19. 19.
    G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 4th ed., Oxford Univ. Press, 1959.Google Scholar
  20. 20.
    T. L. Heath, Diophantus of Alexandria: A Study in the History of Greek Algebra, 2nd ed., Dover, 1964. (Contains a translation into English of Diophantus’ Arithmetica, a 130-page Introduction to Diophantus’ and related work, and a 60-page Supplement on Fermat’s number-theoretic work.)Google Scholar
  21. 21.
    T. L. Heath (ed.), The Thirteen Books of Euclid’s Elements, 3 vols., 2nd ed., Dover, 1956.Google Scholar
  22. 22.
    K. Iga, A dynamical systems proof of Fermat’s little theorem, Math. Mag. 76 (2003) 48-51.CrossRefMathSciNetGoogle Scholar
  23. 23.
    K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, Springer, 1982.Google Scholar
  24. 24.
    I. Kleiner, A History of Abstract Algebra, Birkhäuser, 2007.Google Scholar
  25. 25.
    D. Mackenzie and B. Cipra, What’s Happening in the Mathematical Sciences, Amer. Math. Soc., 2006.Google Scholar
  26. 26.
    M. S. Mahoney, The Mathematical Career of Pierre de Fermat, 2nd ed., Princeton Univ. Press, 1994.MATHGoogle Scholar
  27. 27.
    B. Mazur, Mathematical perspectives, Bull. Amer. Math. Soc. 43 (2006) 309–401.CrossRefGoogle Scholar
  28. 28.
    B. Mazur, Questions about powers of numbers, Notices Amer. Math. Soc. 47 (2000) 195–202.MATHMathSciNetGoogle Scholar
  29. 29.
    L. J. Mordell, Diophantine Equations, Academic Press, 1969.Google Scholar
  30. 30.
    C. J. Mozzochi, The Fermat Diary, Amer. Math. Soc., 2000.Google Scholar
  31. 31.
    H. Riesel, Prime Numbers and Computer Methods for Factorization, 2nd ed., Birkhäuser, 1994.Google Scholar
  32. 32.
    W. Scharlau and H. Opolka, From Fermat to Minkowski: Lectures on the Theory of Numbers and its Historical Development, Springer, 1985.Google Scholar
  33. 33.
    J. Stillwell, Elements of Number Theory, Springer, 2003.Google Scholar
  34. 34.
    J. Stillwell, Mathematics and its History, 2nd ed., Springer, 2002.Google Scholar
  35. 35.
    P. Tannery and Ch. Henry (eds.), Oeuvres de Fermat, 4 vols., Gauthier-Villars, 1891-1912, and a Supplément, ed. by C. de Waard, 1922.Google Scholar
  36. 36.
    C. Vaughan and T. D. Wooley, Waring’s problem: a survey. In Number Theory for the Millennium III, ed. by M. A. Bennett et al, A K Peters, 2002, pp. 301-340.Google Scholar
  37. 37.
    A. Weil, Number Theory: An Approach through History, from Hammurapi to Legendre, Birkhäuser, 1984.Google Scholar
  38. 38.
    H. C. Williams, Solving the Pell equation. In Number Theory for the Millennium III, ed. by M. A. Bennett et al, A K Peters, 2002, pp. 397-435.Google Scholar
  39. 39.
    B. H. Yandell, The Honors Class: Hilbert’s Problems and their Solvers, A K Peters, 2002.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada

Personalised recommendations