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 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
Prime number theorem
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“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 is selected near to some large integer N, the probability that the selected integer is prime is about 1 / ln(N), where ln(N) denotes the naturallogarithm of N. For example, near N = 1,000 about one in seven numbers is prime, whereas near N = 10,000,000,000 about one in 23 numbers is prime. In other words, the average gap between consecutive prime numbers near N is roughly ln(N).
In mathematics, a Mersenne number, named after Marin Mersenne (a French monk who began the study of these numbers in the early 17th century), is a positive integer that is one less than a power of two:
Some definitions of Mersenne numbers require that the exponent p be prime, since the associated number must be composite if p is composite. A Mersenne prime is a Mersenne number that is prime. It is known that if 2p − 1 is prime then p is prime, so it makes no difference which Mersenne number definition is used. As of October 2009, 47 Mersenne primes are known. The largest known prime number (243,112,609 – 1) is a Mersenne prime. Since 1997, all newly-found Mersenne primes have been discovered by the “Great Internet Mersenne Prime Search” (GIMPS), a distributed computing project on the Internet.
The logarithm of a number is the exponent by which another fixed value, the base, has to be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 1000 is 10 to the power 3: 1000 = 103 = 10 × 10 × 10. More generally, if x = by, then y is the logarithm of x to base b, and is written logb(x), solog10(1000) = 3. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations. They were rapidly adopted by scientists, engineers, and others to perform computations more easily and rapidly, using slide rules and logarithm tables. Application of Prime Numbers|
Prime factors are used to find the highest common factor (HCF) and the lowest common multiple (LCM) of two (or more) large numbers. Highest Common Factor (HCF) by Prime FactorsThe HCF of two (or more) numbers is the product of common prime factors. Example 22Find the HCF of 300 and 375.Solution:Organise the above information as shown below and circle the prime factors that are common to both numbers. The HCF is the product of common prime factors. Note: * 300 = 4 × 75 and 375 = 5 × 75. * 75 is the largest factor common to 300 and 375. Lowest Common Multiple (LCM) by Prime FactorsThe LCM of two (or more) numbers is calculated as follows: * Express the numbers as a product of prime factors. * Circle all of the prime factors of the smaller of the two numbers. * Circle any prime factors of a larger number that have not already been circled for the smaller number (or smaller numbers if you are looking for the LCM of more than two numbers). * The LCM is the product of the circled prime factors.
These steps are better understood by reading the following examples. Example 23Find the LCM of 300 and 375.Solution:Organise the above information as shown below and circle all of the prime factors of the smaller number. Then circle any prime factors of the larger number that have not already been circled in the smaller number. The LCM is the product of the circled prime factors. Note: * The product of prime factors of 375 has three 5s but the product of prime factors of 300 has only two 5s. Since we circled two 5s for 300, we must circle the extra 5 for 375 as shown above. * 1500 is a multiple of 300 as 5 × 300 = 1500. * 1500 is a multiple of 375 as 4 × 375 = 1500. Example 24Find the LCM of 6 and 8.Solution:Organise the above information and circle all of the prime factors of the smaller number as shown below. Then circle any prime factors of the larger number that have not already been circled in the smaller number.
The LCM is the product of circled prime factors. Finding a PatternBy observation from Example 24, we find that: In general:The lowest common multiple of two (or more) numbers can be computed as follows: * Express each number as a product of its prime factors using powers. * Then circle the prime factors to the highest power from the given numbers. * Find the product of each prime factor to its highest power from the given numbers. This product represents the LCM of the numbers. Example 25Find the LCM of 9, 40 and 48.Solution:Organise the above information and circle each prime factor with its highest power as shown below. The highest power of 2 is 24, the highest power of 3 is 32 and the highest power of 5 is 51. So, circle 24, 32 and 51. The LCM is the product of the prime factors to the highest powers.|