# Number Theory

## Current Applications

Number theory was labeled the "Queen of Mathe matics" by Gauss. For many years it was thought to be without many practical applications. That situation has changed significantly in the twentieth century with the rise of computers.

Prime and composite numbers play an important role in modern cryptography or coding systems. Huge volumes of confidential information (credit card numbers, bank account numbers) and large amounts of money are transferred electronically around the world every day, all of which must kept secret. One of the important applications of number theory is keeping secrets.

Using Fermat's theorem, a computer can quickly compute if a number-even a large number-is prime. However, once a computer finds out that a number is not prime, it then takes a long time to find out what its factors are, especially if the number is a large composite (say 120 digits long). It can take years on a supercomputer to find the prime factors of large composite numbers.

This time gap between finding out if a number is prime and factoring the primes in a composite number is useful to cryptographers. To create a security system, they invent numerical codes for the letters and characters of a message. Then they use an encoding algorithm (a series of steps to solve a problem) to turn a message into a long number. If the message is more than a certain length, say 100 characters, then the cryptography program breaks the message into blocks of 100 characters. Once the message is translated into a number, the program multiplies the number of an encoded message by a certain prime (which could be a 100 digit number) and by a composite number. The composite number is the product of two prime numbers, which have been randomly selected and which must be in both the encoding algorithm of the sender and the decoding algorithm of the receiver. The prime numbers making up the composite number are usually quite long (100 digits and longer). When the message is transmitted from the sender to the receiver, some of the numbers are made public, but the primes that make up the composite number are kept secret. They are only known by the decoding algorithm of the authorized person who receives the message. Anyone who is eavesdropping on the transmission will see a lot of numbers, but without the prime numbers from the encoding and decoding programs, it is impossible to decode the message in any reasonable time.

Computer cryptography systems are only one application of number theory. Other formulas of number theory allow computer programs to find out many years in advance what days of the week will fall on what dates of the month, so that people can find out well in advance what day of the week Christmas or the Fourth of July will occur. Many computers have preinstalled internal programs that tell users when they last modified a file down to the second, minute, hour, day of the week, and date of the month. These programs work thanks to the formulas of number theorists.

## Resources

### Books

Davenport, Harold. The Higher Arithmetic: An Introduction to the Theory of Numbers. 6th ed. Cambridge: Cambridge University Press, 1992.

Dunham, William. The Mathematical Universe. New York: John Wiley and Sons, 1994.

Peterson Ivars, The Mathematical Tourist: Snapshots of Modern Mathematics. New York: W.H. Freeman and Company, 1988.

Rosen, Kenneth. Elementary Number Theory and Its Applications. 4th ed. Boston: Addison-Wesley, 2000.

Stopple, Jeffrey. A Primer of Analytic Number Theory: From Pythagoras to Riemann. Cambridge: Cambridge University Press, 2003.

Vinogradov, Ivan Matveevich. Elements of Number Theory. Dover Publications, 2003.

Weisstein, Eric W. The CRC Concise Encyclopedia of Mathematics. New York: CRC Press, 1998.

### Periodicals

"Finessing Fermat, Again: The Wily Proof may Finally Be Finished." Scientific American 272.2 (February 1995): 16.

Patrick Moore

## KEY TERMS

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Algorithm

—Method for solving a numerical problem consisting of a series of mathematical steps.

Composite number

—A number that can be divided into two or more prime numbers in addition to 1 and itself.

Congruence

—The relationship between two numbers if they have the same remainder when they are divided by a number.

Cryptography

—The study of creating and breaking secret codes.

Factors

—Numbers that are multiplied with other numbers to equal a product. In the multiplication of 2 × 2 = 4, each 2 is a factor. Also, factors are the numbers that result when numbers are divided. For example, 10 and 10 are factors of 100.

Modulo

—A number by which two other numbers can be divided to give the same remainder.

Prime numbers

—Numbers that can only be divided evenly by 1 and themselves.

Product

—The result of multiplying two or more numbers. Six multiplied by 7 gives the product of 42.

Set

—A set is a collection of things called members or elements of the set. In mathematics, the members of a set will often be numbers.

Whole numbers

—The positive integers: 1, 2, 3, 4...