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.

Why is Juliet Under Pressure in a Scene :: essays research papers

There are two families that hate each other Romeo's family the Montague's and Juliet's family the Capulet's. The families have hated each other for many generations. Romeo and Juliet met at a party even though Lord Capulet has found Juliet a husband but she doesn't like him and falls in love with Romeo who was previously in love with Rosaline. Romeo and Juliet get married in secret hoping in the long run that this deed will end the family feud but Juliet's family don't know about the wedding. Mercutio Romeo's best friend and Tybalt Juliet's cousin get into a fight and Mercutio dies but Romeo turns up and kills Tybalt. Romeo is banished to Mantua for killing Tybalt so Juliet isn't happy because she has lost two of the people she cares about most. At the start of this scene Romeo has sneaked into Juliet?s bedroom and has stopped the night in her bed. In the morning Juliet does not want Romeo to leave as she exclaims ?it was the nightingale, and not the lark.? Which means that the nightingale sounds in the night and the lark sounds in the morning meaning that if it is the lark Romeo must go but Juliet believes it was the nightingale so that Romeo can stay? Romeo knows he must go and pronounces ?it was the lark, the herald of the morn.? Meaning that the lark is the sign of the morning and the nightingale sing?s at night so if it was the lark it would be time for Romeo to go but Juliet can?t stand it when Romeo is gone so she is convinced it was the nightingale. When the nurse comes in and tells Juliet that Romeo must leave because her mother is coming Juliet finally realises that Romeo has got to go and she starts weep for Romeo. When her mother comes in she assumes that Juliet is in mourning for Tybalt, ?Evermore weeping for your cousin?s death.? So Juliet just goes along with it ?feeling so loss, I cannot choose but ever weep the friend.? Lady Capulet tells Juliet her daughter that she will have happy days, ?sudden day of joy,? meaning that amongst all of this madness she will have a happy day. Juliet asks what will happen on this day and when this day is. Lady Capulet. ?Marry, my child, early next Thursday morn.? Juliet is very shocked by this, ?

Ethics of Autonomous Drones in the Military

Jared May Professor Elfstrom February 25, 2013 Intro to Ethics A Soldier, Taking Orders From Its Ethical Judgment Center In this article the author Cornelia Dean has three major points that are supported by arguments made by others. The first major important point is the hopeful idea that autonomous robots can perform more ethically in combat situations than any soldier in the same scenario.She states that even the best and most trained soldiers that are in the midst of battle may not always be able to act accordingly with the battlefield rules of engagement that were stated by the Geneva Convention because of possible lashing out from normal human emotions such as anger, fear, resent, and vengefulness. The second major point Dean wants to show, by the views and studies of others, in her article is that with this possible step in our evolution of military technology we do not want to let this idea fade away.Another major point is if we do develop this technology how would we do so, a nd if not, would we regret not advancing in this field further many years from now. With all of this information Dean uses to present her ideas there are still major flaws such as, the majority of these ideas and beliefs are theoretical, they still have not been fully tested, there is error in all technologies, and where else would the technological advancements lead artificial intelligence.The first argument providing support for Dean’s major point comes from the research hypothesis and thoughts of a computer scientist at Georgia Institute of Technology named Ronald Arkin. Arkin is currently under contract by the United States Army to design software programs for possible battlefield and current battlefield robots. The research hypothesis of Arkin is that he believes that intelligent autonomous robots can perform much more ethically in the heat of the battlefield than humans currently can.Yet this is just a hypothesis and while there is much research done towards this hypoth esis there are still no absolutely positive research information that states an autonomous robot drone can in fact perform better than any soldier on the ground or up in a plane could do. In Arkins hypothesis, he stated that these robots could be designed with no sense of self-preservation. This means that without one of the strongest fears for humans, the fear of death, these robots would be able to understand, compute, and react to situations with out outside extraneous emotions.Although the men and women designing these robot programs may be able to eliminate this psychological problem of scenario fulfillment, which will cause soldiers to retain information that is playing out easier with a bias to pre-existing ideas, it is not always the case that this happens to soldiers. You have to realize that from the second a soldier begins his training he is trained and taught to eliminate the sense of self-preservation. There are isolated incidents with soldier error, but they are and wi ll be corrected by superior officers or their fellow soldiers.Another factor that affects Cornelia Dean’s arguments is that there are errors in all things including technology. Throughout history there have been new uses of technology in warfare but with these come problems and error flaws that have cause and can cause more casualties than needed. With the use of an Automated drone the belief by Dean is that it will be able to decide whether or not to launch an attack on a high priority target whether or not if the target is in a public are and will decide if the civilian casualties would be worth it.But what happens if that drone is only identifying the target and the number of civilians surrounding it? It will not be able to factor in what type of people would be around him such as men, women, or children and any variance of them. The error in this situation would be the drone saying the target is high enough priority and a missile is launched and the civilians were women a nd children around while a school bus was driving by.The casualties would then instantly out weigh the priority to eliminate a specific target and a human pilot would much easier abort a mission than a predetermined response of an autonomous robot. Although Ronald Arkin believes there are situations that could arise when there may not be time for a robotic device to relay back what is happening to a human operator and wait for how to respond in the situation that could complete a mission, it may be that second of time delay between the robot and human operator that the ethical judgment is made.Also the realization that many robots in which are operated by humans are widely used to detect mines, dispose of or collects bombs, and clear out buildings to help ensure extra safety of our soldiers is a way that robots are already used today as battlefield assistants supports Dean. But all of these machines in the field have moments of failure or error. When the machines do fail it takes a soldier who has trained for that experience to fix and then use it again. If an autonomous drone fails while on a mission it is completely by its self and no human operator to fix it.Then can arise the problem of enemies realizing they were even being monitored and they could gain access to our military technology and can eventually use it against us. Another major point that Cornelia Dean discusses upon is with this possible step in our evolution of military technology we do not want to let this idea fade away. A large part of that is if we do develop this technology how would we do so, and if not, how much would we regret or how much would it affect us for not advancing in this field further many years from now.The argument that if other countries advance upon this faster and better than the United States military we could become less of a world power and be more at risk of attack and war with greater human fatalities is not necessarily true. This situation is important in the sen se of keeping up with the other world powers but I believe that the risk for reward is not worth the amount of damage and civilian casualties that could happen from any number of robotic drones and their possible errors.There is a possibility as the technology develops and robots become more and more aware to the point were, Arkin believes that, they can make decisions at a higher level of technological development. Yet if these autonomous robots truly can think for themselves and make decisions brings a whole new possibility of problems of what if the robot can decide something differently than what the developers originally had programmed. Also comes the actual use problem of can the government ethically accept that in early stages of use, even after extraneous testing, there may be accidental casualties.If a robot has any error of making decisions because of how new and un-tested they are any of the possibly terrible results would not be the responsibility of the robot but of the country and government that designed it. The supporting evidence of this article strongly shows that Cornelia Dean will hope that use of these ethically superior autonomous robots will be apart of our military in the near future before the United States fall behind to other super powers in the world.Yet with all of this information Dean uses to present her ideas there are still major flaws such as, the majority of these ideas and beliefs are theoretical, they still have not been fully tested, and that there is error in all technologies. With these major points being enforced with plenty of evidence throughout the article, and with all of the possible negative sides and errors of this argument, it is safe to say that this will be and is a controversial topic of discussion by many governments and all parties involved with this technological advancement.

The Treaty Of Versailles And Its Effect On The World War I

The Treaty of Versailles The Treaty of Versailles was a treaty signed between Germany and the Allies which consisted of Britain, France and America. The idea of the Treaty was to end World War one and Germany would be too weak to start another war. This meant that there would be peace throughout Europe for a long time but it was controversial at best. It was signed in the Versailles palace which was large enough for hundreds of people to be involved in the signing on 28th of June 1919. Germany had almost no say in what was put into the treaty but they had no choice but to sign it otherwise they would be invaded by the Allies. â€Å"Its 15 parts and 440 articles reassigned German boundaries and assigned liability for reparations†Ã¢â‚¬ ¦show more content†¦Along with the reduction of men, â€Å"Germany was not allowed tanks, submarines or military aircraft and the navy could only have six battleships† (Mr Hinds’ History; what were the main Terms of the Versailles Treaty 2016). This was created so that Germany would not have the force to declare war but still had the power to stop communism from getting into Western Europe. Another aspect to the Treaty of Versailles were the economic terms. These terms stated that Germany had to pay 6.6 billion dollars back to the allies. Germany was not able to pay this amount of money so they had to borrow money from nearby countries. This meant that Germany fell into huge amounts of debt and lost their title for second most economically advanced country in the world. There were two more important Terms for The Treaty of Versailles, they were the General terms and the territorial terms. There were three important general clauses to the treaty. The first was that Germany had to take all responsibility for the war, this was called the War Guilt Clause and it was the term that Germany was the most upset about. The second was that Germany had to pay for all war damages which was mostly to France and Belgium. The third was that a League of Nations was set up to keep world peace and Germany was not allowed to join. There were also the territorial terms of the treaty where Germany had to give up