Topics in a Number Theory Course

To learn more about a topic listed below, click the topic name to go to the corresponding MathWorld classroom page.

Congruence	An equation in modular arithmetic, i.e., one in which only the remainders relative to some base, known as the "modulus," are significant.
Continued Fraction	A real number as a nested fraction, useful in number theory.
Convergent	(1) A partial sum of a continued fraction. (2) Tending towards some definite finite value.
Diophantine Equation	An equation for which only integer solutions are allowed.
Divisor Function	(1) The number of distinct divisors of a given integer. (2) The sum of some fixed power of the divisors.
Elliptic Curve	A curve defined by an irreducible cubic polynomial in two variables.
Euclidean Algorithm	An algorithm for finding the greatest common divisor of two numbers.
Euler-Mascheroni Constant	The limit of the difference between the nth partial sum of the harmonic series and the natural logarithm of n, approximately equal to 0.577.
Fermat's Last Theorem	A famous discovery by Fermat that took 350 years to prove; it states that any number that is a power greater than two cannot be the sum of two like powers.
Number Theory	A field of mathematics sometimes called "higher arithmetic" consisting of the study of the properties of integers. Primes and prime factorization are especially important concepts in number theory.
Partition	A way of writing a whole number as a sum of positive integers in which the order of the addends is not significant.
Perfect Number	A positive integer that equals the sum of its divisors.
Prime Counting Function	Given x, the prime counting function returns the number of primes less than or equal to x.
Prime Factorization Algorithms	The many algorithms that have been devised for determining the prime factors of a given number (a process called prime factorization).
Prime Number Theorem	A theorem specifying frequency of prime numbers.
Quadratic Reciprocity Theorem	A theorem that tells whether a quadratic equation modulo a prime has a solution.
Riemann Zeta Function	A special function of mathematics and physics that is intimately related to deep results surrounding the prime number theorem.
Squarefree	A positive integer is squarefree if it is not divisible by any perfect square greater than one.
Totient Function	The number of positive integers less than or equal to a given number that are relatively prime to it.
Transcendental Number	A number that is not the root of any polynomial with integer coefficients. The opposite of algebraic number. MathWorld