Prime Numbers and Applications
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Prime number
A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering. This theoremrequires excluding 1 as a prime.
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Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of theintegers. Number theorists study prime numbers (which, when multiplied, give all the integers) as well as the properties of objects made out of integers (such as rational numbers) or defined as generalizations of the integers (such as, for example, algebraic integers).
Integers can be considered either in themselves or as solutions to equations (diophantine geometry). Questions in number theory are often best understood through the study ofanalytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter
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Prime number theorem
From Wikipedia, the free encyclopedia
"PNT" redirects here. For other uses, see PNT (disambiguation).
In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers.
Informally speaking, the prime number theorem states that if a random integer
Prime number
A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering. This theoremrequires excluding 1 as a prime.
-------------------------------------------------
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of theintegers. Number theorists study prime numbers (which, when multiplied, give all the integers) as well as the properties of objects made out of integers (such as rational numbers) or defined as generalizations of the integers (such as, for example, algebraic integers).
Integers can be considered either in themselves or as solutions to equations (diophantine geometry). Questions in number theory are often best understood through the study ofanalytical objects (e.g., the Riemann zeta function) that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, e.g., as approximated by the latter
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Prime number theorem
From Wikipedia, the free encyclopedia
"PNT" redirects here. For other uses, see PNT (disambiguation).
In number theory, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers.
Informally speaking, the prime number theorem states that if a random integer