number theory, branch of mathematics concerned with the properties of the integers (the numbers 0, 1, −1, 2, −2, 3, −3, …). An important area in number theory is the analysis of prime numbers. A prime number is an integer p>1 divisible only by 1 and p; the first few primes are 2, 3, 5, 7, 11, 13, 17, and 19. Integers that have other divisors are called composite; examples are 4, 6, 8, 9, 10, 12, … . The fundamental theorem of arithmetic, the unique factorization theorem, asserts that any positive integer a is a product (a = p₁ · p₂ · p₃ · · · p_n) of primes that are unique except for the order in which they are listed; e.g., the number 20 is the product 20 = 2 · 2 ·5, and it is unique (disregarding order) since 20 has this and only this product of primes. This theorem was known to the Greek mathematician Euclid, who proved that there are infinitely many primes. Analytic number theory has given a further refinement of Euclid's theorem by determining a function that measures how densely the primes are distributed among all integers. Twin primes are primes having a difference of 2, such as (3,5) and (11,13). The modern theory of numbers made its first great advances through the work of Leonhard Euler Euler, Leonhard (lā`ônhärt oi`lər), 1707–83, Swiss mathematician.
..... Click the link for more information. , C. F. Gauss Gauss, Carl Friedrich (kärl frē`drĭkh gous)
..... Click the link for more information. , and Pierre de Fermat Fermat's Last Theorem, which states that the equation xⁿ + yⁿ = zⁿ, where x, y, z, and n are nonzero integers, has no solutions for n that are greater than 2.
..... Click the link for more information. . It remains a major area of mathematical research, to which the most sophisticated mathematical tools have been applied.

Bibliography

See O. Ore, Number Theory and Its History (1988); R. P. Burn, A Pathway into Number Theory (2d ed. 1996); J. H. Silverman, A Friendly Introduction to Number Theory (1996); M. A. Herkommer, Number Theory: A Programmer's Guide (1998); R. A. Mollin, Algebraic Number Theory (1999).

number theory

Branch of mathematics concerned with properties of and relations among integers. It is a popular subject among amateur mathematicians and students because of the wealth of seemingly simple problems that can be posed. Answers are much harder to come up with. It has been said that any unsolved mathematical problem of any interest more than a century old belongs to number theory. One of the best examples, recently solved, is Fermat's last theorem.

number theory [′nəm·bər ′thē·ə·rē]

(mathematics)

The study of integers and relations between them.