Prime Numbers Divide

Prime poem game DividePrime Numbers History, Facts and ExamplesPrime Numbers An IntroductionPrime subroutine is the compute, which is greater than 1 and undersurfacenot be come apartd by any publication excluding itself and one. A tiptop egress is a positive(p) integer that has just two positive integer factors, including 1 and itself. Such as, if the factors of 28 atomic number 18 listed, there are 6 factors that are 1, 2, 4, 7, 14, and 28. Similarly, if the factors of 29 are listed, there are only two factors that are 1 and 29. Therefore, it plunder be inferred that 29 is a pinnacle number, but 28 is not.Examples of peak comeThe first few prime numbers are as follows2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, etc.Identifying the primesThe ancient Sieve of Eratosthenes is a simple way to work out each(prenomina l) prime numbers up to a given limit by preparing a list of each(prenominal) integers and repetitively striking out multiples of already lay down primes. There is also a modern Sieve of Atkin, which is more complex when compared to that of Eratosthenes.A method to determine whether a number is prime or not, is to divide it by all primes less than or equal to the square root of that number. If the results of any of the divisions are an integer, the original number is not a prime and if not, it is a prime. One need not actually calculate the square root once one sees that the quotient is less than the divisor, one can stop. This is called as the trial division, which is the simplest primality mental render but it is impractical for testing large integers because the number of possible factors grows exponentially as the number of digits in the number to be tested increases.Primality tests A primality test algorithm is an algorithm that is used to test a number for primality, that i s, whether the number is a prime number or not.AKS primality testThe AKS primality test is based upon the comparing(x a)n = (xn a) (mod n) for a coprime to n, which is received if and only if n is prime. This is a generalization of Fermats little theorem extended to polynomials and can easily be proven development the binomial theorem unitedly with the fact that for all 0 (x a)n = (xn a) (mod n, x r 1), which can be checked in polynomial time.Fermat primality testFermats little theorem asserts that if p is prime and 1 a a p -1 1 (mod p)In order to test whether p is a prime number or not, one can scavenge random as in the interval and check if there is an equality.Solovay-Strassen primality testFor a prime number p and any integer a, A (p -1)/2 (a/p) (mod p)Where (a/p) is the Legendre symbol. The Jacobi symbol is a generalisation of the Legendre symbol to (a/n) where n can be any odd integer. The Jacobi symbol can be computed in time O((log n)) using Jacobis generalization of law of quadratic reciprocity.It can be observed whether or not the congruenceA (n -1)/2 (a/n) (mod n) holds for various values of a. This congruence is true for all as if n is a prime number. (Solovay, Robert M. and Volker Strassen, 1977)Lucas-Lehmer test This test is for a natural number n and in this test, it is also required that the prime factors of n 1 should be already cognize.If for every(prenominal) prime factor (q) of n 1, there exists an integer a less than n and greater than 1 such asa n -1 1 (mod n)and thena n -1/q 1 (mod n) then n is prime. If no such number can be found, n is complex number.Miller-Rabin primality testIf we can find an a such thatad 1 (mod n), and a2nd -1 (mod n) for all 0 r s 1then a proves the compositeness of n. If not, a is called a strong liar, and n is a strong probable prime to the base a. Strong liar refers to the case where n is composite but yet the equations hold as they would for a prime number.There are several witnesses a for every odd composite n. But, a simple way to generate such an a is known. Making the test probabilistic is the solution we choose randomly, and check whether it is a witness for the composite spirit of n. If n is composite, majority of the as are witnesses, therefore the test will discover n as a composite number with high probability. (Rabin, 1980)A probable prime is an integer, which is considered to be probably prime by passing a certain test. Probable primes, which are actually composite (such as Carmichael numbers) are known as pseudoprimes.Besides these methods, there are separate methods also. There is a set of Diophantine equations in 9 variables and one logical argument in which the parameter is a prime number only if the resultant system of equations has a solution over the natural numbers. A single formula with the property of all the positive values being prime can be obtained with this method. There is another formula that is based on Wilsons theorem. The number two is generated several generation and all other primes are generated exactly once. Also, there are other similar formulas that can generate primes. Some primes are categorized as per the properties of their digits in decimal or other bases. An example is that the numbers whose digits develop a palindromic sequence are palindromic primes, and if by consecutively removing the first digit at the go forth or the right generates only new prime numbers, a prime number is known as a truncatable prime.The first 5,000 prime numbers can be known very quickly by just looking at odd numbers and checking each new number (say 5) against every number above it (3) so if 5Mod3 = 0 then its not a prime number.History of prime numbersThe most ancient and acknowledged proof for the debate that There are infinitely many prime numbers, is given by Euclid in his Elements (Book IX, Proposition 20). The Sieve of Eratosthenes is a simple, ancient algorithm to key all prime numbers up to a particular intege r. After this, came the modern Sieve of Atkin, which is faster but more complex. The Sieve of Eratosthenes was created in the third century BC by Eratosthenes. Some clues can be found in the surviving records of the ancient Egyptians regarding their knowledge of prime numbers for example, the Egyptian constituent expansions in the Rhind papyrus befool fairly different forms for primes and for composites. But, the first surviving records of the clear consider of prime numbers come from the Ancient Greeks. Euclids Elements (circa 300 BC) allow in key theorems about primes, counting the fundamental theorem of arithmetic and the infinitude of primes. Euclid also explained how a perfect number is constructed from a Mersenne prime.After the Greeks, nothing special happened with the study of prime numbers till the 17th century. In 1640, Pierre de Fermat affirmed Fermats little theorem, which was later on proved by Leibniz and Euler. Chinese may have identified a special case of Fermats theorem much earlier. Fermat assumed that all numbers of the form 22n + 1 are prime and he proved this up to n = 4. But, the subsequent Fermat number 232+1 is composite whose one prime factor is 641). This was later on discovered by Euler and now no raise Fermat numbers are recognized as prime numbers. A French monk, Marin Mersenne looked at primes of the form 2p 1, with p as a prime number. They are known as Mersenne primes after his name.Euler showed that the infinite series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + is divergent. In 1747, Euler demonstrated that even the perfect numbers are in particular the integers of the form 2p-1(2p-1), where the second factor is a Mersenne prime. It is supposed that there are no odd perfect numbers, but it is not proved yet. In the beginning of the 19th century, Legendre and Gauss independently assumed that because x tends to infinity, the number of primes up to x is asymptotic to x/log(x), where log(x) is the natural logarithm of x.Awards for findi ng primesA prize of US$100,000 has been offered by the Electronic Frontier Foundation (EFF) to the first discoverers of a prime with a minimum 10 million digits. Also, $150,000 for 100 million digits, and $250,000 for 1 billion digits has been offered. In 2000, $50,000 for 1 million digits were paid. Apart from this, prizes up to US$200,000 for finding the prime factors of particular semi-primes of up to 2048 bits were offered by the RSA Factoring Challenge.Facts about prime numbers73939133 is an amazing prime number. If the last or the digit at the units place is removed, every time you will get a prime number. It is the largest known prime with this property. Because, all the numbers which we get after removing the end digit of the number are also prime numbers. They are as follows 7393913, 739391, 73939, 7393, 739, 73 and 7. All these numbers are prime numbers. This is a distinct quality of the number 73939133, which any other number does not have. (Amazing number facts, 2008) Th e only even prime number is 2. All other even numbers can be divided up by 2. So, they are not prime numbers. Zero and 1 are not considered to be prime numbers. If the sum of the digits of a number is a multiple of 3, that number can be divided by 3.With the exception of 0 and 1, a number is either a prime number or a composite number. A composite number is identified as any number that is greater than 1 and that is not prime. The last digit of a prime number greater than 5 can never be 5. Any number greater than 5 whose last digit is 5 can be divided by 5. (Prime Numbers, 2008) 1/20.5 Terminates 1/30.33333Repeating seal off up 1 digit 1/50.2 Terminates 1/70.1428571428Repeating block 6 digits 1/110.090909Repeating block 2 digits 1/130.0769230769Repeating block 6 digits 1/170.05882352941176470588Repeating block 16 digits 1/190.0526315789473684210526Repeating block 18 digits 1/230.04347826086956521739130434Repeating block 22 digits For some of the prime numbers, the size of the it erate block is 1 less than the prime.These are known as Golden Primes.2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 979 primes out of the 25 (less than 100) are golden primes this forms 36% (9/25). (Amazing number facts, 2008)Examples of mathematicians specialized in prime numbers Arthur Wieferich, D. D. Wall, Zhi Hong Sun and Zhi Wei Sun, Joseph Wolstenholme, Joseph Wolstenholme, Euclid, Eratosthenes.Applications of prime numbers For a long time, the number theory and the study of prime numbers as well was seen as the canonical example of pure mathematics with no applications beyond the self-interest of studying the topic. But, in the 1970s, it was publicly announced that prime numbers could be used as a basis for creating the public key cryptography algorithms. They were also used for hash tables and pseudorandom number generators.A number of rotor coil machines were designed with a different number of pins on each rotor. The number of pins on any one rotor was either prime, or co-prime to the number of pins on any other rotor. With this, a full cycle of possible rotor positions (before repeating any position) was generated.Prime numbers in the arts and literatureAlso, prime numbers have had a significant influence on several artists and writers. The French composer Olivier Messiaen created ametrical music through natural phenomena with the use of prime numbers. In his works, La Nativit du Seigneur (1935) and Quatre tudes de rythme (1949-50), he has used motifs with lengths given by different prime numbers to create unpredictable rhythms 41, 43, 47 and 53 are the primes that appear in one of the tudes. A scientist of NASA, Carl Sagan recommended (in his science fiction Contact) that prime numbers could be used for communication with the aliens. The award-winning play Arcadia by Tom Stoppard was a headstrong attempt made to discuss mathematical ideas on the stage. In the very first scene, the 13 year old heroine baffles over the Fermats last theorem (theorem that involves prime numbers). A popular fascination with the mysteries of prime numbers and cryptography has been seen in various films.ReferencesAmazing number facts, 2008. Retrieved April 28, 2008 from http//www.madras.fife.sch.uk/maths/amazingnofacts/fact018.html Prime Numbers, 2008. Retrieved April 28, 2008 from http//www.factmonster.com/ipka/A0876084.html Solovay, Robert M. Strassen, V. (1977). A fast Monte-Carlo test for primality. SIAM journal on Computing 6 (1) 84-85.Rabin, M.O. (1980). Probabilistic algorithm for testing primality, Journal of Number Theory 12, no. 1, pp. 128-138.