Introduction to
Number Theory

Session Transcript: Introduction to Number Theory

Number theory is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."

Fundamental to number theory is the concept of divisibility. For integers a and b, b is divisible by a (written a | b) if there exists an integer k such that b = ak. Primes are the building blocks of integers. A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The Fundamental Theorem of Arithmetic states that every integer greater than 1 either is a prime number itself or can be represented as the product of prime numbers and that this representation is unique, up to the order of the factors.

Modular arithmetic, often called "clock arithmetic," is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. Two integers a and b are said to be congruent modulo n if their difference a − b is an integer multiple of n.