Number theory 1998

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Encyclopaedia of Mathematical Sciences Free Preview. Buy eBook. Buy Hardcover. Buy Softcover. FAQ Policy. About this book This book is a survey of the most important directions of research in transcendental number theory. Show all. Show next xx. Read this book on SpringerLink.

Recommended for you. Parshin I. Shafarevich Translated by Koblitz, N. 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, The modern theory of numbers made its first great advances through the work of Leonhard Euler , C.

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

Number theory

See O. Burn, A Pathway into Number Theory 2d ed. Mollin, Algebraic Number Theory 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.

Number theory a branch of mathematics that studies the properties and relationships of numbers. Specifically, it deals with the natural, or counting numbers, including prime numbers. Number theory is important because the simple sequence of counting numbers from one to infinity conceals many relationships beneath its surface.

One of the most important distinctions in number theory is between prime and composite numbers. Prime numbers can only be divided evenly with nothing left over by 1 and themselves. Prime numbers include 2, 3, 5, 7, 11, 13, 17, and so on to infinity.


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The number 1 is not considered a prime. All primes are odd numbers except for 2, because any even number can be divided evenly by 2. A composite number can be divided, or factored, into two or more prime numbers in addition to 1 and itself. Ten is a composite number because it can be divided by 2, 5, 1, and itself. The numbers 2 and 5 are the prime factors of Any whole number that is not a prime is a composite. One difference between prime and composite numbers is that it takes relatively little time to determine if a number is prime, but far longer to determine the prime factors of a composite number, especially if the composite is very large digits or more.

This discrepancy in computation time is important in developing computer security systems. Prime numbers do not occur in a predictable way. There are sequences of primes which can be partially described in a formula, but sooner or later the formula breaks down.

One formula, invented by French philosopher and mathematician Marin Mersenne — is 2 p - 1, where p is a prime number. Although this formula generates many primes, it also misses many primes. Another formula, invented by Swiss mathematician and physicist Leonhard Euler — , generates prime numbers regularly for the series of consecutive numbers from 0 to 15 and then stops. Number theory is full of famous formulas that illustrate the relationships between whole numbers from 1 to infinity. Some of these formulas are very complicated, but the most famous ones are very simple, for example, the theorem by Fermat below that proves if a number is prime.

French mathematician Pierre de Fermat — is one of the most famous number theoreticians in history, but mathematics was only his hobby. Fermat was a judge in France , and he published very little during his life. He did correspond extensively with many leading intellectuals of his day, and his mathematical innovations were presented to these pen pals in his letters.

Say n is any whole number , and p is any prime number. Raise n to the power of p, and then subtract n from the result. If p is really a prime number, then the result can be divided evenly by p.

If anything is left over after the division, then the number p is not prime. A shorter way of putting this formula is this: n p - n can be divided evenly by p. Subtract 8 from the result is Divide by 3: the result is German mathematician Karl Friedrich Gauss — made many contributions to pure and applied mathematics. He was born to poor parents in Germany. His high intelligence was noticed early and nurtured by his mother and uncle, but his father never encouraged Gauss in his education. In effect, his theory of congruence allows people to break up the infinite series of whole numbers into smaller, more manageable chunks of numbers and perform computations upon them.

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This arrangement makes the everyday arithmetic involved in such things as telling time much easier to program into computers. Gauss said that if one number is subtracted from another a - b , and the remainder of the subtraction can be divided by another number, m, then a and b are congruent to each other by the number m. The remainder, , can be divided by 60, 20, 10, and other numbers. However, for this purpose, only 60 will be applied. That is, both and are related to a third number the remainder after is subtracted from , which can be multiplied by In an abstract sense, this computation is related to such everyday arithmetic functions as telling the time of day on a digital watch.

If the watch says it is in the afternoon, then, from one frame of reference, it has subtracted minutes from the minute period between 12 noon to midnight.

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What remains is minutes past noon. The remainder, , can be divided evenly by the modulo 60 and by other numbers which will be ignored. Clocks are already divided into modulos and one simply notes the hour and how many minutes come before or after the hour. Number theory is an immensely rich area and it is defined by the important problems that it tries to solve. Sometimes a problem was considered solved, but years later the solution was found to be flawed. One important challenge in number theory has been trying to find a formula that will describe all the prime numbers.

To date, that problem has not been solved. Two of the most famous problems in number theory involve Fermat. Many mathematicians, including Mersenne and Euler, have tried to find a formula that will define all the prime numbers. No one has ever succeeded. Fermat had one of the most famous failures. He thought that if he squared 2 and then raised the square of 2 to a higher power, which he labeled n a whole number , then the results would be nothing but primes.

This formula appeared to work until Leonhard Euler proved it wrong.