Number Theory

A vast and fascinating field of mathematics consisting of the study of the properties of whole numbers. Primes and Prime Factorization are especially important in number theory, as are a number of functions such as the Divisor Function, Riemann Zeta Function, and Totient Function. Excellent introductions to number theory may be found in Ore (1988) and Beiler (1966). The classic history on the subject (now slightly dated) is that of Dickson (1952).

See also Arithmetic, Congruence, Diophantine Equation, Divisor Function, Gödel's Incompleteness Theorem, Peano's Axioms, Prime Counting Function, Prime Factorization, Prime Number, Quadratic Reciprocity Theorem, Riemann Zeta Function, Totient Function

References

Number Theory

Andrews, G. E. Number Theory. New York: Dover, 1994.

Andrews, G. E.; Berndt, B. C.; and Rankin, R. A. (Ed.). Ramanujan Revisited: Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, June 1-5, 1987. Boston, MA: Academic Press, 1988.

Apostol, T. M. Introduction to Analytic Number Theory. New York: Springer-Verlag, 1976.

Ayoub, R. G. An Introduction to the Analytic Theory of Numbers. Providence, RI: Amer. Math. Soc., 1963.

Beiler, A. H. Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, 2nd ed. New York: Dover, 1966.

Bellman, R. E. Analytic Number Theory: An Introduction. Reading, MA: Benjamin/Cummings, 1980.

Berndt, B. C. Ramanujan's Notebooks, Part I. New York: Springer-Verlag, 1985.

Berndt, B. C. Ramanujan's Notebooks, Part II. New York: Springer-Verlag, 1988.

Berndt, B. C. Ramanujan's Notebooks, Part III. New York: Springer-Verlag, 1997.

Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1993.

Berndt, B. C. Ramanujan's Notebooks, Part V. New York: Springer-Verlag, 1997.

Berndt, B. C. and Rankin, R. A. Ramanujan: Letters and Commentary. Providence, RI: Amer. Math. Soc, 1995.

Borwein, J. M. and Borwein, P. B. Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.

Brown, K. S. ``Number Theory.'' http://www.seanet.com/~ksbrown/inumber.htm.

Burr, S. A. The Unreasonable Effectiveness of Number Theory. Providence, RI: Amer. Math. Soc., 1992.

Burton, D. M. Elementary Number Theory, 4th ed. Boston, MA: Allyn and Bacon, 1989.

Carmichael, R. D. The Theory of Numbers, and Diophantine Analysis. New York: Dover, 1959.

Cohn, H. Advanced Number Theory. New York: Dover, 1980.

Courant, R. and Robbins, H. ``The Theory of Numbers.'' Supplement to Ch. 1 in What is Mathematics?: An Elementary Approach to Ideas and Methods, 2nd ed. Oxford, England: Oxford University Press, pp. 21-51, 1996.

Davenport, H. The Higher Arithmetic: An Introduction to the Theory of Numbers, 6th ed. Cambridge, England: Cambridge University Press, 1992.

Davenport, H. and Montgomery, H. L. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, 1980.

Dickson, L. E. History of the Theory of Numbers, 3 vols. New York: Chelsea, 1952.

Dudley, U. Elementary Number Theory. San Francisco, CA: W. H. Freeman, 1978.

Friedberg, R. An Adventurer's Guide to Number Theory. New York: Dover, 1994.

Gauss, C. F. Disquisitiones Arithmeticae. New Haven, CT: Yale University Press, 1966.

Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994.

Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, 1979.

Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1959.

Hasse, H. Number Theory. Berlin: Springer-Verlag, 1980.

Ireland, K. F. and Rosen, M. I. A Classical Introduction to Modern Number Theory, 2nd ed. New York: Springer-Verlag, 1995.

Klee, V. and Wagon, S. Old and New Unsolved Problems in Plane Geometry and Number Theory. Washington, DC: Math. Assoc. Amer., 1991.

Koblitz, N. A Course in Number Theory and Cryptography. New York: Springer-Verlag, 1987.

Landau, E. Elementary Number Theory, 2nd ed. New York: Chelsea, 1988.

Lang, S. Algebraic Number Theory, 2nd ed. New York: Springer-Verlag, 1994.

Lenstra, H. W. and Tijdeman, R. (Eds.). Computational Methods in Number Theory, 2 vols. Amsterdam: Mathematisch Centrum, 1982.

LeVeque, W. J. Fundamentals of Number Theory. New York: Dover, 1996.

Mitrinovic, D. S. and Sandor, J. Handbook of Number Theory. Dordrecht, Netherlands: Kluwer, 1995.

Niven, I. M.; Zuckerman, H. S.; and Montgomery, H. L. An Introduction to the Theory of Numbers, 5th ed. New York: Wiley, 1991.

Ogilvy, C. S. and Anderson, J. T. Excursions in Number Theory. New York: Dover, 1988.

Ore, Ø. Invitation to Number Theory. Washington, DC: Math. Assoc. Amer., 1967.

Ore, Ø. Number Theory and Its History. New York: Dover, 1988.

Rose, H. E. A Course in Number Theory, 2nd ed. Oxford, England: Clarendon Press, 1995.

Rosen, K. H. Elementary Number Theory and Its Applications, 3rd ed. Reading, MA: Addison-Wesley, 1993.

Schroeder, M. R. Number Theory in Science and Communication: With Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, 3rd ed. New York: Springer-Verlag, 1997.

Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, 1993.

Sierpinski, W. 250 Problems in Elementary Number Theory. New York: American Elsevier, 1970.

Uspensky, J. V. and Heaslet, M. A. Elementary Number Theory. New York: McGraw-Hill, 1939.

Vinogradov, I. M. Elements of Number Theory, 5th rev. ed. New York: Dover, 1954.

Weil, A. Basic Number Theory, 3rd ed. Berlin: Springer-Verlag, 1995.

Weil, A. Number Theory: An Approach Through History From Hammurapi to Legendre. Boston, MA: Birkhäuser, 1984.

Weyl, H. Algebraic Theory of Numbers. Princeton, NJ: Princeton University Press, 1998.