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Number Theory

Number theory

Number theory is the study of natural numbers. Natural numbers are the counting numbers that we use in everyday life: 1, 2, 3, 4, 5, and so on. Zero (0) is often considered to be a natural number as well.

Number theory grew out of various scholars' fascination with numbers. An example of an early problem in number theory was the nature of prime numbers. A prime number is one that can be divided exactly only by itself and 1. Thus 2 is a prime number because it can be divided only by itself (2) and by 1. By comparison, 4 is not a prime number. It can be divided by some number other than itself (that number is 2) and 1. A number that is not prime, like 4, is called a composite number.

The Greek mathematician Euclid (c. 325270 b.c.) raised a number of questions about the nature of prime numbers as early as the third century b.c. Primes are of interest to mathematicians, for one reason: because they occur in no predictable sequence. The first 20 primes, for example, are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, and 71. Knowing this sequence, would you be able to predict the next prime number? (It is 73.) Or if you knew that the sequence of primes farther on is 853, 857, 859, 863, and 877, could you predict the next prime? (It is 883.)

Questions like this one have intrigued mathematicians for over 2,000 years. This interest is not based on any practical application the answers may have. They fascinate mathematicians simply because they are engrossing puzzles.

Famous theorems and problems

Studies in number theory over the centuries have produced interesting insights into the properties of natural numbers and ongoing puzzles about such numbers. As just one example of the former, consider Fermat's theorem, a discovery made by French mathematician Pierre de Fermat (16011665). Fermat found a quick and easy way to find out if a particular number is a prime or composite number. According to Fermat's theorem, one can determine if any number (call that number p ) is a prime number by the following method: choose any number (call that number n ) and raise that number to p. Then subtract n from that calculation. Finally, divide that answer by p. If the division comes out evenly, with no remainder, then p is a prime number.

Fermat was also responsible for one of the most famous puzzles in mathematics, his last theorem. This theorem concerns equations of the general form xn + yn = zn. When n is 2, a very familiar equation results: x2 + y2 = z2, the Pythagorean equation of right-angled triangles.

The question that had puzzled mathematicians for many years, however, was whether equations in which n is greater than 2 have any solution. That is, are there solutions for equations such as x3 + y3 = z3, x4 + y4 = z4, and x5 + y5 = z5? In the late 1630s, Fermat wrote a brief note in the margin of a book saying that he had found proof that such equations had no solution when n is greater than 2. He never wrote out that proof, however, and for more than three centuries mathematicians tried to confirm his theory.

As it turns out, any proof that Fermat had discovered was almost certainly wrong. In 1994, Princeton University professor Andrew J. Wiles announced that he had found a solution to Fermat's theorem. But flaws were soon discovered in Wiles's proof (which required more than 150 pages of mathematical equations). By late 1994 Wiles thought the flaws had been solved. However, it will take several years before other mathematicians will be able to verify Wiles's work.

Words to Know

Composite number: A number that can be factored into two or more prime numbers in addition to 1 and itself.

Cryptography: The study of creating and breaking secret codes.

Factors: Two or more numbers that can be multiplied to equal a product.

Prime number: Any number that can be divided evenly only by itself and 1.

Product: The number produced by multiplying two or more numbers.

Applications

As mentioned above, the charm of number theory for mathematicians has little or nothing to do with its possible applications in everyday life. Still, such applications do appear from time to time. One such application has come about in the field of cryptographythe writing and deciphering of secret messages (or ciphers). In the 1980s, a number of cryptographers almost simultaneously announced that they had found methods of writing ciphers in such a way that they could be sent across public channels while still remaining secrets. Those methods are based on the fact that it is relatively easy to raise a prime number to some exponent but very difficult to find the prime factors of a large number.

For example, it is relatively simple, if somewhat time-consuming, to find 358143. Actually, the problem is not even time-consuming if a computer is used. However, finding the prime factors of a number such as 384,119,982,448,028 is very difficult unless one knows one of the prime factors to begin with. The way public key cryptography works, then, is to attach some large number, such as 384,119,982,448,028, as a "key" to a secret message. The sender and receiver of the secret message must know one of the prime factors of that number that allows them to decipher the message. In theory, any third party could also decipher the message provided that they could figure out the prime factors of the key. That calculation is theoretically possible although, in practice, it takes thousands or millions of calculations and a number of years, even with the most powerful computers now known.

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number theory

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 · · · pn) 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, C. F. Gauss, and Pierre de Fermat. It remains a major area of mathematical research, to which the most sophisticated mathematical tools have been applied.

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).

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number theory

number theory Branch of mathematics concerned with the properties of natural numbers (whole numbers) or special classes of natural numbers such as prime numbers and perfect numbers. The 4th-century bc Greek mathematician Euclid proved that the number of primes was infinite. One of the unresolved problems in number theory is to find formulae for the generation of the primes. Fermat (in the 17th century) and Euler (in the 18th century) both explored number theory.

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