The Road Not Taken By Robert Frost Essay Example For Students

The Road Not Taken By Robert Frost Essay Robert Frost is the most well known American poet who draws on nature as a subject for his poems. Nearly all of his poetry can be related to the outdoors and a free feeling that Frost seems to cherish. This also was reflected in his life as he lived around nature for the majority of his life. It is not hard to see through his poems how deeply moved he is by the Earth and the miracle of life. He did suffer through an array of hardships during his life, but still kept an appreciation for what he still had. Robert Lee Frost was born in San Francisco on March 26, 1874. His father, William was a Harvard educated journalist who worked for the Daily Evening Post and was elected as a delegate to the Democratic National Convention in 1880. Isabelle, his mother moved to America at the age of 12 as the daughter of a Scottish sea captain. His mother was his primary source of education and home schooled him for much of his early life. His father passed away in 1884 from tuberculosis and his mothe r moved him and his sister to Lawrence, Massachusetts. This is the first time that Robert is really close to nature and he begins his love for the outdoors here. The basis of our government was realized by Lord Acton, a British historian when he wrote, Power tends to corrupt, and absolute power corrupts absolutely. He knew that if any one person or group ran a country, they would soon become power crazed and lose the respect and support of its citizens. This is the reason why our forefathers came up with a system of checks and balances to ensure that no one group could control the entire government. Lord Acton was not the first to believe in a separated government. Philosophers dating back to Aristotle favored a government that contained the elements of monarchy, aristocracy, and democracy. John Locke later wrote that the best way to eliminate corruption in government was to separate the powers of the legislative and executive branches. Montesquieu added the powers of the judiciary branch to complete what we now call separation of powers. Our forefathers only had to refine the philosophers ideas to come up with our present system of government. Basically they wrote that the executive branch enforces the laws, the legislative branch passes the laws, and the judiciary branch interprets the laws. The president runs the executive branch. The president has many powers including the right to veto a bill. If Congress passes a bill, the president can veto it and unless each house has a two-thirds vote to override the veto, the bill never becomes a law. Over the history of our government, presidents have vetoed over 2,500 acts of Congress and been overridden over 100 times. The president can also call Congress into a special session if they do not act upon proposed legislation by the president. Since the president is head of a political party he can easily influence Congress into legislation. The president can appeal directly to the public in order to influence Congres s. Presidents also have a unique power to pardon people who have been convicted of federal crimes. Another task of our president is to begin the process of appointing all federal judges and other officers of the government, such as his cabinet, ambassadors, ministers, and consuls. Before these positions can be appointed Congress also has to approve each of the officers who were elected by the president. All Supreme Court Judges have been elected by the president then approved by Congress before being appointed. The relationships of the United States and other foreign countries are mostly determined by the president He has the power to decide whether to recognize new nations and governments, and in turn to negotiate treaties with them. He cannot however do this alone. He is dependent on the Senate to approve the treaties with a two-thirds vote. The Senate does not always need to approve the negotiations with foreign nations. The president can negotiate executive agreements without ha ving to be approved by the Senate. .u24a55148d687b6599dad61417bf19bf3 , .u24a55148d687b6599dad61417bf19bf3 .postImageUrl , .u24a55148d687b6599dad61417bf19bf3 .centered-text-area { min-height: 80px; position: relative; } .u24a55148d687b6599dad61417bf19bf3 , .u24a55148d687b6599dad61417bf19bf3:hover , .u24a55148d687b6599dad61417bf19bf3:visited , .u24a55148d687b6599dad61417bf19bf3:active { border:0!important; } .u24a55148d687b6599dad61417bf19bf3 .clearfix:after { content: ""; display: table; clear: both; } .u24a55148d687b6599dad61417bf19bf3 { display: block; transition: background-color 250ms; webkit-transition: background-color 250ms; width: 100%; opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #95A5A6; } .u24a55148d687b6599dad61417bf19bf3:active , .u24a55148d687b6599dad61417bf19bf3:hover { opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #2C3E50; } .u24a55148d687b6599dad61417bf19bf3 .centered-text-area { width: 100%; position: relative ; } .u24a55148d687b6599dad61417bf19bf3 .ctaText { border-bottom: 0 solid #fff; color: #2980B9; font-size: 16px; font-weight: bold; margin: 0; padding: 0; text-decoration: underline; } .u24a55148d687b6599dad61417bf19bf3 .postTitle { color: #FFFFFF; font-size: 16px; font-weight: 600; margin: 0; padding: 0; width: 100%; } .u24a55148d687b6599dad61417bf19bf3 .ctaButton { background-color: #7F8C8D!important; color: #2980B9; border: none; border-radius: 3px; box-shadow: none; font-size: 14px; font-weight: bold; line-height: 26px; moz-border-radius: 3px; text-align: center; text-decoration: none; text-shadow: none; width: 80px; min-height: 80px; background: url(https://artscolumbia.org/wp-content/plugins/intelly-related-posts/assets/images/simple-arrow.png)no-repeat; position: absolute; right: 0; top: 0; } .u24a55148d687b6599dad61417bf19bf3:hover .ctaButton { background-color: #34495E!important; } .u24a55148d687b6599dad61417bf19bf3 .centered-text { display: table; height: 80px; padding-left : 18px; top: 0; } .u24a55148d687b6599dad61417bf19bf3 .u24a55148d687b6599dad61417bf19bf3-content { display: table-cell; margin: 0; padding: 0; padding-right: 108px; position: relative; vertical-align: middle; width: 100%; } .u24a55148d687b6599dad61417bf19bf3:after { content: ""; display: block; clear: both; } READ: Herman Melville And Moby Dick Essay We will write a custom essay on The Road Not Taken By Robert Frost specifically for you for only $16.38 $13.9/page Order now The Congress, which runs the legislative branch is made up of two houses, the Senate and the House of Representatives. The Senate contains two officials elected from each of the fifty states and population determines the number of officials in the House. This means that the more densely populated states have a higher number of representatives. Congress is where amendments and bills are first voted on. In the case of the president vetoing their proposed bill, the Congress can vote again and with a two-third majority still pass the bill. It then passes through to the Supreme Court, which is the last stop for a proposed bill. It is there that the judges determine whether the new bill is Unconstitutional and if so the bill is not passed. In all 155 congressional acts have been deemed by the Supreme Court to be Unconstitutional. The federal budget has to be approved by the Congress. All revenue bills must originate in the House then be approved by the Senate. This is to ensure that the St ates with larger populations cannot control the money situation. In fact any bill that is passed by one house can be voted against in the other. With a majority, that bill will not be passed through for the presidents approval. The Senate also has a few powers that are reserved strictly for them. Only the Senate has to approve the presidents choice for government officials and ambassadors. Only nine cabinet nominees have ever been denied by the Senate. They are the only house needed to ratify treaties that are proposed by the president. In the matter of impeachment both houses have powers that they can call their own. The House has to bring charges of misconduct and then the Senate tries the cases and determines whether they are guilty or innocent. Both houses of Congress have shared responsibility for economic decisions including taxes, borrowing money, regulating commerce between states and with foreign countries, making money and stating its value, punishing counterfeiters, and d etermining bankruptcy laws. In addition the houses determine the rules and regulations for the naturalization of foreign citizens, set the standard for weights and measures, provide for post offices and public roads, issue patents and copyrights, punish piracy, and establish federal courts. They also provide for an army and a navy and can declare war. They can use these military forces in order to uphold our laws. The final branch of our government is called the judiciary branch. It is made up of both the federal and state court systems. There were state courts long before the Constitutional Convention, which brought up a debate on whether we should even have a federal court system. They soon decided that we should keep the state courts and add a federal court system with limited power. The first Congress then divided the country into districts and created a federal court for each one. The federal court system now consists of the Supreme Court, 91 district courts, 11 appeals courts, and three courts of special jurisdiction. In order to become a federal justice one has to be nominated by the president then approved by the Congress. Judges are not elected for a certain number of years; instead they hold their positions during good behavior. This simply means that until they die, retire or resign they remain in office. If a judge ever commits an offense they can be impeached by Congress and lose their job. Congress carries even more power over the federal court system. Congress can create or abolish any federal court besides the Supreme Court. It also determines how many judges are used in the judiciary system and what their pay scale is. .ue4b618f73c6b64edc90ed7efa5ba0cd1 , .ue4b618f73c6b64edc90ed7efa5ba0cd1 .postImageUrl , .ue4b618f73c6b64edc90ed7efa5ba0cd1 .centered-text-area { min-height: 80px; position: relative; } .ue4b618f73c6b64edc90ed7efa5ba0cd1 , .ue4b618f73c6b64edc90ed7efa5ba0cd1:hover , .ue4b618f73c6b64edc90ed7efa5ba0cd1:visited , .ue4b618f73c6b64edc90ed7efa5ba0cd1:active { border:0!important; } .ue4b618f73c6b64edc90ed7efa5ba0cd1 .clearfix:after { content: ""; display: table; clear: both; } .ue4b618f73c6b64edc90ed7efa5ba0cd1 { display: block; transition: background-color 250ms; webkit-transition: background-color 250ms; width: 100%; opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #95A5A6; } .ue4b618f73c6b64edc90ed7efa5ba0cd1:active , .ue4b618f73c6b64edc90ed7efa5ba0cd1:hover { opacity: 1; transition: opacity 250ms; webkit-transition: opacity 250ms; background-color: #2C3E50; } .ue4b618f73c6b64edc90ed7efa5ba0cd1 .centered-text-area { width: 100%; position: relative ; } .ue4b618f73c6b64edc90ed7efa5ba0cd1 .ctaText { border-bottom: 0 solid #fff; color: #2980B9; font-size: 16px; font-weight: bold; margin: 0; padding: 0; text-decoration: underline; } .ue4b618f73c6b64edc90ed7efa5ba0cd1 .postTitle { color: #FFFFFF; font-size: 16px; font-weight: 600; margin: 0; padding: 0; width: 100%; } .ue4b618f73c6b64edc90ed7efa5ba0cd1 .ctaButton { background-color: #7F8C8D!important; color: #2980B9; border: none; border-radius: 3px; box-shadow: none; font-size: 14px; font-weight: bold; line-height: 26px; moz-border-radius: 3px; text-align: center; text-decoration: none; text-shadow: none; width: 80px; min-height: 80px; background: url(https://artscolumbia.org/wp-content/plugins/intelly-related-posts/assets/images/simple-arrow.png)no-repeat; position: absolute; right: 0; top: 0; } .ue4b618f73c6b64edc90ed7efa5ba0cd1:hover .ctaButton { background-color: #34495E!important; } .ue4b618f73c6b64edc90ed7efa5ba0cd1 .centered-text { display: table; height: 80px; padding-left : 18px; top: 0; } .ue4b618f73c6b64edc90ed7efa5ba0cd1 .ue4b618f73c6b64edc90ed7efa5ba0cd1-content { display: table-cell; margin: 0; padding: 0; padding-right: 108px; position: relative; vertical-align: middle; width: 100%; } .ue4b618f73c6b64edc90ed7efa5ba0cd1:after { content: ""; display: block; clear: both; } READ: Scarlet Letter And Scaffold EssayJudicial power is used in many different sorts of cases. A court is used to settle disputes in any case which arises under the Constitution, the laws and treaties of the United States, and any which affect ambassadors of foreign countries in America. They attend to any maritime cases and pretty much any controversy which includes the government. They are also responsible for any controversy between states and a foreign nation. That includes American citizens that have disagreements with a citizen of another country. However, they cannot hear cases with a citizen against a state government as well as any cases concerning individual state law s. Aside from trying cases the Supreme Court is also the final step

The Search for Meaning in Slaughterhouse Five essays

The Search for Meaning in Slaughterhouse Five essays Kurt Vonnegut's novel Slaughterhouse Five represents a man's desperate, yet, useless search for meaning in a senseless existence. Vonnegut uses a narrator, which is different from the main character to develop his theme. Vonnegut introduces Slaughterhouse Five in first person point of view. In the second chapter, however, this narrator changes to a bystander who speaks from a third person perspective. Vonnegut wants the reader to realize that the narrator and Billy Pilgrim, the main character, are two different people. In order to do this, Vonnegut places the narrator in the text, multiple times. An American near Billy wailed that he had excreted everything but his brains...That was I. That was me.? This statement clearly illustrates that the narrator and Billy are not the same people. The narrator was the American disgusted by Billy. Vonnegut places his experiences and his views in the text. He begins the book by stating, "All this happened, more or less." The war parts, anyway, are pretty much true...I?ve changed all of the names.? He feels war is a senseless act and, Slaughterhouse Five allows Vonnegut to express his feelings on the matter. Through Billy Pilgrim, he is able to portray his views. They had been lying in ambush for the Germans. They had been discovered and shot from behind. Now they were dying in the snow, feeling nothing, turning the snow the color of raspberry sherbet. So it goes.? He uses vivid and meaningful imagery here. The reader can picture the snow slowly being colored with the blood of the soldiers. By ending with the statement,So it goes,? The reader is satisfied. Vonnegut uses this statement throughout the book to show that death is death, there is no glorious or great death; all death is equal. Vonnegut doesn't want to glorify war. The narrator made a vow to O?Hare's wife , in chapter one, that the story would not do this. ...I give my word of honor. I'll call it the children's crusade.? ...

Personal Development Essay Example | Topics and Well Written Essays - 1250 words

Personal Development - Essay Example In addition, some level of qualitative research will be conducted to determine real-life attitudes of currently-practicing nurses and administration (where appropriate) to determine what credentials or habits would be best-suited to an administrative nursing role. I believe these practices will enhance my knowledge of the nursing profession by being able to link theoretical nursing theory with tangible practice dynamics. As part of my vision for excellence in holistic nursing practice, servant leadership and transformational leadership will be critical to becoming a well-rounded nursing professional able to take a solid leadership position in a dynamic health environment. Servant leadership demands having a focus on others, providing empathetic and benevolence toward others while still developing my own professional competencies (Farazmand et al., 2010). Servant leadership demands being altruistic in virtually dimensions of practice, creating a global perspective that recognizes dive rsity of culture and non-biased servitude for a variety of different patients and professionals. Transformational leadership is also critical for becoming a Chief Nursing Officer as this requires the ability to establish followership, team performance and loyalty from subordinates. Fairholm (2009) describes transformational leadership as being inspirational, while setting a vision for team practice, and then routinely conveying this vision through role modeling and constant communication with team members. Transformational leadership requires the nurse to be a teacher and coach, helping others to develop their own competencies and, ultimately, self-actualization at the psychological level. Transformational leadership provides local-level leadership within the microeconomic health care environment. I must also be considerate of my own personal needs as a professional in this field. I have conducted several self-analyses regarding my preferred learning styles, personality type and lea dership characteristics. These assessments and evaluations have returned very consistent results that indicate I am well-suited for a servant leadership role, maintaining considerable empathetic characteristics and high emotional intelligence. An effective leader in a highly-visible nursing role in administration must be able to gauge the emotional states and needs of their followers and patients in order to provide effective care. At the same time, I must be equipped to regulate my own emotional responses in professional meeting environments, when dealing with disheartening patient scenarios, and when working with diverse cultural representatives. Personal satisfaction will come by finally being self-actualized in relation to my very legitimate desire to assist others in need. Personal satisfaction will also be achieved by having a dominant role in the organization, which tends to suit many of my personality characteristics. Based on honest self-assessment and the results of many d ifferent evaluation tests, I have many dominant characteristics when it comes to leadership, both socially and professionally. I am very self-confident about my abilities and my problem-solving competencies and therefore I trust in my

Writer's Chioce Movie Review Example | Topics and Well Written Essays - 1250 words

Writer's Chioce - Movie Review Example For instance, the movie depicts the overflowing of Manhattan as a wind-caused surge, even though this epic wave, which is also known as tsunami could only be stimulated by a marine earthquake or tremor or a meteorite strike, rather than by a hurricane or an alteration in the North Atlantic Current. The stun frosting in the view of the hurricane goes against the laws of thermodynamics (Rehill, 2009). The filmmakers are honest about the truth that it is not a precisely realistic scenario. The promotion material given by the film firm Fox says that once scientists converse about sudden climate change they imply ten or five years, but for theatrical reasons, the whole thing was condensed to a couple of weeks. In order to depict the dramatic impacts of a vital climatic catastrophe in a short time span, they only took acknowledged weather limits such as storm surges, tornados, hailstorms and cyclones blizzards. Conversely, given the constraints and rules of the genre, it is astonishing to what level the filmmakers have attempted to include some sensible background. For instance, in the early moments of the film, the director reveals a United Nations climate convention in Delhi where Jack Hall talks about the probable risk of a blackout of the North Atlantic Current (Ebert, 2005). During the convention, Jack Hall states that a blackout may occur in a thousand years or hundred years, or may not at all. Several real climate experts have said a similar thing. In this manner, the actor shows what climate experts think in a realistic way in the movie. The director of the movie shows the politics of weather change. It is disturbingly realistic how the leader of the United States delegation, who is the vice president in the movie, reacts to Halls speech. This implies that small prospects with few words of dialogue are smartly used to initiate a number of key conflicts and ideas, which are extremely recognizable to climatologists but not

Mongolia Tourism Article Essay Example | Topics and Well Written Essays - 1250 words

Mongolia Tourism Article - Essay Example The outcomes of the study were utilized in making recommendations to the Mongolian tourism authority on efficient targeting of its international tourism market and improvement of tourism services in the country. 2. A number of factors have influenced the growth of international tourism in Mongolia. The collapse of the eastern European communism system, poor economic conditions, forest fires and adverse weather conditions made international tourism in Mongolia to deteriorate between 1990 and 1997. Business activities brought tourists from China and Russia Federation across Mongolia and this led to a sharp improvement in Mongolian international tourism in 1998. Other factors that improved tourism in Mongolia include favorable visa regulations that enabled tourists to get visas at the Mongolian border. The National Tourism Board that was promulgated in 1995 helped market Mongolia as a favorable tourism destination. Businesspeople from Korea and Japan who worked in Mongolia spread news back in their home countries about Mongolia. The internet helped raise awareness about Mongolia, increased accessibility to travel advertising and reservations and made them less costly. The good impression that Mongolia created in its early Japanese and Korean visitors was another factor in the improvement of tourism in the country (Yu & Goulden, 2006). 3. Cognitive dimension is one of the constructs through which tourism satisfaction can be viewed. The dimension entails tourists’ experience with services. Tourists’ reaction to service performance is captured in the affective dimension of tourism. Systemic dimension of tourism articulates the disparity between the services that tourists expect and things that they get while on the ground. The cross-cultural dimensions of tourism emphasize the impact that cultural difference has on tourists’ perception of service delivery and quality. The dimension reckons that the difference between the

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...