Number Theory

Number theory is a vast and fascinating field of mathematics, sometimes called "higher arithmetic," 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 (2005abc).

The great difficulty in proving relatively simple results in number theory prompted no less an authority than Gauss to remark that "it is just this which gives the higher arithmetic that magical charm which has made it the favorite science of the greatest mathematicians, not to mention its inexhaustible wealth, wherein it so greatly surpasses other parts of mathematics." Gauss, often known as the "prince of mathematics," called mathematics the "queen of the sciences" and considered number theory the "queen of mathematics" (Beiler 1966, Goldman 1997).

REFERENCES:

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.

Anglin, W. S. The Queen of Mathematics: An Introduction to Number Theory. Dordrecht, Netherlands: Kluwer, 1995.

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, 1997a.

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, 1997b.

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 & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.

Bressoud, D. M. and Wagon, S. A Course in Computational Number Theory. London: Springer-Verlag, 2000.

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.

Cohen, H. Advanced Topics in Computational Number Theory. New York: Springer-Verlag, 2000.

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, Vol. 1: Divisibility and Primality. New York: Dover, 2005a.

Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, 2005b.

Dickson, L. E. History of the Theory of Numbers, Vol. 3: Quadratic and Higher Forms. New York: Dover, 2005c.

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.