Prime Numbers Divide

Prime Numbers Divide Prime Numbers: History, Facts and Examples Prime Numbers: An Introduction Prime number is the number, which is greater than 1 and cannot be divided by any number excluding itself and one. A prime number is a positive integer that has just two positive integer factors, including 1 and itself. Such as, if the factors of 28 are 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 can be inferred that 29 is a prime number, but 28 is not. Examples of prime numbers The first few prime numbers are as follows: 2, 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 primes The ancient Sieve of Eratosthenes is a simple way to work out all prime numbers up to a given limit by preparing a list of all integers and repetitively striking out multiples of already found 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 test 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 is, whether the number is a prime number or not. AKS primality test The AKS primality test is based upon the equivalence (x a)n = (xn a) (mod n) for a coprime to n, which is true if and only if n is prime. This is a generalization of Fermats little theorem extended to polynomials and can easily be proven using the binomial theorem together 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 test Fermats 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 pick random as in the interval and check if there is an equality. Solovay-Strassen primality test For 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) ²) using Jacobis generalization of law of quadratic reciprocity. It can be observed whether or not the congruence A (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 known. If for every prime factor (q) of n − 1, there exists an integer a less than n and greater than 1 such as a n -1 ≠¡1 (mod n) and then a n -1/q 1 (mod n) then n is prime. If no such number can be found, n is composite number. Miller-Rabin primality test If we can find an a such that ad ≠¡ 1 (mod n), and a2nd -1 (mod n) for all 0 ≠¤ r ≠¤ s 1 then ‘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. â€Å"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 nature 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 other methods also. There is a set of Diophantine equations in 9 variables and one parameter 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 times 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 left 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 numbers The most ancient and acknowledged proof for the statement that â€Å"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 identify all prime numbers up to a particular integer. After this, came the modern Sieve of Atkin, which is faster but more complex. The Sieve of Eratosthenes was created in the 3rd 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 fraction expansions in the Rhind papyrus have fairly different forms for primes and for composites. But, the first surviving records of the clear study of prime numbers come from the Ancient Greeks. Euclids Elements (circa 300 BC) include key theorems about primes, counting the fundamental theorem of arithmetic and the infinitude of primes. Euclid also explained how a perfect number is constructed fro m 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 further 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 finding primes A 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 numbers 73939133 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) The only even prime number is 2. All other even numbers can be divided 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) 1/2 0.5 Terminates 1/3 0.33333 Repeating block: 1 digit 1/5 0.2 Terminates 1/7 0.1428571428 Repeating block: 6 digits 1/11 0.090909 Repeating block: 2 digits 1/13 0.0769230769 Repeating block: 6 digits 1/17 0.05882352941176470588 Repeating block: 16 digits 1/19 0.0526315789473684210526 Repeating block: 18 digits 1/23 0.04347826086956521739130434 Repeating block: 22 digits For some of the prime numbers, the size of the repeating 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, 97 9 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 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 literature Also, 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 Natività © du Seigneur (1935) and Quatre à ©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 à ©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 willful 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. References Amazing 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.

Interviewing the Local Police Essay -- essays research papers

Interviewing the Local Police My independent project was done on a whimsical basis. It's thanksgiving eve and my family and I are all gathered around watching football. The Redskins and Cowboy's are all tied up, and my uncle is on the verge of having a nervous breakdown. A diehard Cowboys fan, who can't even remember when was the last time he didn't bet on a game. Mom and dad are still eating, while my aunt recites a thanksgiving song for all the uninvited guests. The door bell rings, and what do you know it's the local Police. Officers Bob Jacob and William Gould stop by on their neighbor-hood patrol. My aunt invities them in for some coffee, and they end up eating the rest of our thanksgiving dinner. For some strange reason I think of Sociology.( Do you think they'll arrest me if I ask...

Role of Women in Forest Management

————————————————- Cudia, Jane Victoria A. February 23, 2011 2007-15891Soc Sci 180 Role of Women in Forest Management Increasingly, the role of indigenous peoples in forest management and conservation had been recognized on the international level. Even if forest management and conservation had been an indigenous practice since time immemorial, recognition of indigenous peoples roles started only three or four decades ago. However, indigenous peoples as protectors of the environment are taken as a whole disregarding the contribution of roles and differences as contributing factors to the continuous practice of forest management and conservation. In terms of IP roles in forest management and conservation, gender is a very important thing to consider. Given an indigenous social and political context, the management of forests is communal. In general, forest management is primarily the domain of men. Women’s participation on the other hand primarily lies in forest protection because there is an intimate relationship between IP women and the forest (Caguioa, 2011). However, problem lies in the lack of recognition in national policies as to the roles of IP women in forest management and conservation. The pine forest of Brgy. Agawa, Besao, Mt. Province is a communal forest shared by different communities managed mostly by indigenous peoples. For the people of Brgy. Agawa, the forest or langdas is the source of livelihood and a place where their unique culture thrive, one of the major features of indigenous peoples. In terms of livelihood, the langdas is the source of wild fruits and animals, lumber, and firewood. In the indigenous law they practice, selling of pine lumber is prohibited. Also, outsiders are prohibited from getting anything from the langdas making the practice sustainable. In terms of tradition and culture, they believe that there are spirits guarding the rivers and forests. This is one of the reasons why indigenous peoples do not exploit the forest resources. Also, they manage the forest in a sustainable way because of the belief that their ancestors, who were buried in sacred places, mingle with their affairs. The role of women in environment protection, forest management, and conservation is very significant. Two of the key informants of the study conducted by Caguioa (2011) and her colleagues are old women who spent most of their life living in the area. The people of Brgy. Agawa, has a history of resistance in protecting the langdas and the environment. One sign of protest they had done before was the exposure of older women’s breast to oppose people who wanted to operate saw mills in the area back in the 1940s. In response to the secret resin tapping activities done to pine trees that operated during the 1970s, women of Besao secretly removed the plastic catchers and burned all of it. In general, women of Agawa, Besao, Mt. Province show their protest in the regional and national level in opposition of road construction, mining, and logging projects that will ruin the langdas. Amidst globalization, vulgar consumerism, high demands from the market, and laws that treat us all equals [sometimes even without regard to culture], the people of Brgy. Agawa, Besao, Mt. Province especially women managed the forest in a sustainable way. Following their traditions and belief systems, they had managed to conserve the forest by following natural mechanisms to restore the forest. Given the resources and knowledge systems we have as members of the dominant and so called â€Å"developed† society, we should devise forest management and conservation mechanisms that are easy and applicable. However, due to a market-driven economy we engage in, we have no control over our resources anymore. The working mechanism that works today is â€Å"what the market demands, the market gets† even at the expense of the environment and the people who manages to protect and conserve the forest. Forest management, although primarily dominated by men, it should be the domain of all even if there are differences in gender. As seen in the case of Brgy. Agawa, Mt. Province, women had great contributions in forest management and conservation. It all goes down to this: in effective forest management and conservation, gender roles and differences have a lot to offer. Reference: Caguioa, M. C. (2011). Panagsalaknib ti Langdas: Role of Indigenous Women in Forest Management in Brgy. Agawa, Besao, Mt. Province from the Global Lecture Series on Indigenous Peoples’ Studies (University of the Philippines Baguio).

The gap between the rich and the poor today essays

The gap between the rich and the poor today essays The gap between rich and poor in the world today. In the world of 1995, there are still huge differeces between rich and poor, developed and less developed countries. But why? Who is to blame? What can we do about it? Many things have been tried out to solve these problems, but does it work? It seems bizarre, that we, modern, intelligent people, have not yet succeded to get rid of the differences between DCs (developed countries) and LDCs (less developed countries). We try, don't we? Every year, we grant 2% of our Gross National Product, GNP, to foreign aid to help the LDCs to get a better standard of living (better agriculture, more and better schools and hospitals, access to health personell, medicines, etc.). On the other hand, is our "standard of living" the best for LDCs, and the one we should impose on them? For instance, what is the point of giving complex macinery like tractors and harvesters, which need expensive fuel and maintenance, to people who have harvested their crops by manpower for hundreds of years? We know for a fact that the money we grant is not being used adequately. A lot of the money is taken by the governments of the less developed countries, and a great amount of the sum are not being used to the purposes they are meant for. Bribery and corruption are huge problems in developing countries. It makes more sense to dig wells for people who walk for miles every day to get their daily water supply, than to support officials with BMWs and grand houses. The World Bank was established, and a large amound of capital was poured in, despite of the fact that the Third World lacked the level of infrastructure, the economic and social background, and the skilled personnel of Europe. The failure of this model of economic development to produce economic well-being and growth for most Third World countries is due to a number of factors. These factors include the concentration of economic resources in the...

Free Essays on Teenage Runaways

Teenage Runaways How many times did you vow as a teenager that you would run away when your parents wouldn’t allow you to go out? Allow me to paint a picture for you. You’re packing your bags while grumbling about how tough life is at your age. Pink hair, black nails and a nose ring are all that matters in high school. Meanwhile, your parents are calmly standing at the door watching you with concerned yet confident looks knowing that the driveway, maybe even the mailbox, will be your farthest attempt. However, what if you went to a shelter? What if you ended up on the streets? Is your local city bench better than a warm bed? Now, let us replace those loving parents with angry, screaming and abusive ones. Would the situation change and seem more comprehensible? For some central Florida teens the mean streets of Orlando and other large nationwide cities are much more welcoming than their very own home. However, why do young teens run away? Where do most go? And what are city committees doing about this rising epidemic? A child who runs away usually has left home to escape or avoid an unpleasant environment. Most motives in teenagers for running away range from escaping recurrent abusive experiences at home to self improvement where they hope to change or stop whatever negative activity they are doing or about to do ( Conner par. 3). If, as a parent, you are unaware of your child’s friends and who they have close relations and contact with on a regular basis, than the likeliness of your child running away is mounting. An ever-present distance between child and parent is one of the many warning signs in troubled and potential runaways ( Conner par. 7). In addition to unfamiliarity in a child’s network of friends, abusive, irrational and emotional behavior are also key indicators. In this day and age as a parent one may think it impossible to know everyone your child hangs out with or comes in contact with. I... Free Essays on Teenage Runaways Free Essays on Teenage Runaways Teenage Runaways How many times did you vow as a teenager that you would run away when your parents wouldn’t allow you to go out? Allow me to paint a picture for you. You’re packing your bags while grumbling about how tough life is at your age. Pink hair, black nails and a nose ring are all that matters in high school. Meanwhile, your parents are calmly standing at the door watching you with concerned yet confident looks knowing that the driveway, maybe even the mailbox, will be your farthest attempt. However, what if you went to a shelter? What if you ended up on the streets? Is your local city bench better than a warm bed? Now, let us replace those loving parents with angry, screaming and abusive ones. Would the situation change and seem more comprehensible? For some central Florida teens the mean streets of Orlando and other large nationwide cities are much more welcoming than their very own home. However, why do young teens run away? Where do most go? And what are city committees doing about this rising epidemic? A child who runs away usually has left home to escape or avoid an unpleasant environment. Most motives in teenagers for running away range from escaping recurrent abusive experiences at home to self improvement where they hope to change or stop whatever negative activity they are doing or about to do ( Conner par. 3). If, as a parent, you are unaware of your child’s friends and who they have close relations and contact with on a regular basis, than the likeliness of your child running away is mounting. An ever-present distance between child and parent is one of the many warning signs in troubled and potential runaways ( Conner par. 7). In addition to unfamiliarity in a child’s network of friends, abusive, irrational and emotional behavior are also key indicators. In this day and age as a parent one may think it impossible to know everyone your child hangs out with or comes in contact with. I...

Descartes Essay Example | Topics and Well Written Essays - 250 words - 3

Descartes - Essay Example because it is not possible to differentiate the experiences that we have while awake, and the experiences that we have while dreaming; for Descartes, it is possible that we are dreaming while thinking that we are awake. For this reason, therefore, the doubts that Descartes suggests in the dream argument are different and more extreme than the doubts that Descartes suggests in the senses argument; while in the sense argument Descartes argues that it is only some knowledge derived from the senses that can be doubted, in the dream argument Descartes argued and demonstrated that all sensory knowledge can be doubted. The evil demon argument is more extreme than either the sense or the dream argument. This is because, first, while the sense argument suggest that it is only some types of sensory knowledge that can be doubted, the evil demon argument suggest and demonstrate that all sensory knowledge can be doubted. Secondly, while the dream argument demonstrate that sensory knowledge can be doubted, it can reasonably be objected that some simple truths like the truths of mathematics and geometry cannot be doubted; on the other hand, the evil demon argument suggests that even the truths of mathematics and geometry can be doubted because these truths may be nothing but deceptions of an evil genius. While the evil demon argument can deceive Descartes in all his beliefs, the evil demon, however, cannot deceive Descartes into believing that he doesn’t exist. This is because for Descartes, existence is a pre-condition of doubting; according to Descartes, one should exist before doubting. For this reason, Descartes argued that, since he is able to doubt, this means that he exists. Descartes, therefore, concluded that the evil demon cannot deceive him into believing that he doesn’t exist. The main difficulty for Descartes’ philosophical project is accounting for how human body and human mind interact. In giving explanation to how human mind and human body interact,

Consequences of the Peloponnesian War Research Paper

Consequences of the Peloponnesian War - Research Paper Example Although this observation by Thucydides lacked the advantage of hindsight, his statement now carries validity, as the Peloponnesian War had many immediate and lasting effects, which this paper will attempt to determine. In order to properly understand the consequences of the Peloponnesian War, the causes and course of the war must be known. In Donald Kagan’s On the Origins of War and Preservation of Peace, he argues that the causes of all war are sourced from â€Å"fear, honor, and interest† (On the Origins 6), and this holds true with the Peloponnesian War. Athens and Sparta were two of the most powerful Greek city-states in the 5th century B.C., and they were on opposite sides of the â€Å"power bloc† due to the formation of the Delian League and the Peloponnesian League. The Delian League eventually became the Athenian Empire, was originally made to combat the threat of the Persian Empire (The Outbreak 2); the Peloponnesian League was formed by Sparta to comba t the rising threat of Athens (Thucydides, Hammond, Rhodes 476). Rather than combining their respective power and influence, the two city states became opposed factions within the Hellenic World. While there are many intricate and underlying causes to the Peloponnesian War, Thucydides and numerous modern historians agree, to the best of their knowledge, that the main cause of the war was â€Å"Spartan fear of Athenian power† (Thucydides, Hammond, Rhodes 477). Because of the threat of the growing Athenian Empire, in hindsight it became evident that the war was inevitable – the Athenian power became an object of fear that the Spartans could not ignore. The Peloponnesian War spanned across a period of twenty seven years, encompassing numerous theaters, battles and campaigns that cannot be explained entirely in this paper. This paper will outline a brief summary of the war that will be used to help determine the consequences of the conflict. Athens was aware of the fact th at they could not outright defeat the Spartan army, thus, they built a walled corridor between their city and their port of Piraeus, which the Athenians resided within in an attempt to wait out the Spartan army and outlast them in a war of attrition (Daniel 74). Since the Spartans could not breach the walls of their enemy and the Athenians could not outlast the Spartans, the war resulted to a series of Athenian naval raids and Spartan attacks into Athenian land with the goal of destroying vital crops and resources (Daniel 74). After a plague within the Athenian walls that led to the death of the Athenian war leader Pericles, Alcibiades, a new Athenian leader, took the reins of the Athenian forces, and drastically altered the Athenian plan of action for the war. Alcibiades decided to change from a defensive strategy to an offensive one, and thus ordered an invasion of the city of Syracuse on the island of Sicily, which, due to bad leadership, organization, and excellent Spartan defen se, turned out to be a failure (Daniel 75). The failed campaign resulted in the destruction of the Athenian fleet and army, and ultimately resulted in the Athenians losing the war that they had started (Gombrich 63). First, this paper will analyze the immediate effects of the war on both Athens and Sparta. As history has seen in numerous instances, being defeated in a war has seemingly endless and perpetual