Letter Free Essays

Douglas M. Stillwell Vice President, Human Resources Memorial Health Center Stockton Falls, Ohio 43210 Dear Mr. Stillwell: Please accept my application for the Assistant Administrator position that appeared in this month`s ACHE Job Bank. We will write a custom essay sample on Letter or any similar topic only for you Order Now Recently I served as an Administrator in Training for Marion House Health Care Center (add where that is). As my attached resume describes, I worked with that organization from October 2005 to October 2006. I am prepared to assume the greater management challenges at a larger health care facility like yours.   My experience during my association with Marion House Health Care Center, included planning, implementing and evaluating many of the facility`s programs and activities. I reported my observations and activities to my direct supervisor, the facility administrator. Two of my academic courses proved especially useful while I served as administrator in training. The HSA 571 Medical Informatics Masters course gave me the knowledge and skills necessary to participate in planning, implementing, managing, and evaluating health care information systems for the facility. HSA 685 Special Topics: Quality Cost Consumer Issues in Health Care Masters course helped me develop a better awareness of the concepts and emerging issues in quality, cost containment and the growth of consumerism in health care. Thank you for all your time and consideration. Your listed position offers an ideal fit with my professional background and career goals. I would appreciate an opportunity to interview for the position. I will telephone at the end of the week to discuss this opportunity further. Sincerely, Tejesh Patel, MHSA How to cite Letter, Essay examples Letter Free Essays I have already uploaded our family pictures on our new Website. Practically took the time segregating and scanning the images by page and by year. I was also thinking about your move to run as Governor while browsing the site. We will write a custom essay sample on Letter or any similar topic only for you Order Now There are good things and bad things associated with its duties and responsibilities but I firmly feel in my heart that you can overcome all the bad things that come with the position especially during the campaign. I have reared you to be a brave man of honor. Run as you see fit and serve your country, your countrymen and our God well. Always remember to practice accountability. Make it your personal policy to keep a detailed statement of your expenditures. When your separate accounting software is ready I can guarantee you my full support in tracking and monitoring expenditures, people and results. You need to personally check the flow of your transactions, money and the results to avoid sabotage. Keep your antennae out for information to safeguard your integrity in carrying out public duties. It is always a good thing to be transparent and honest. Treat your friends and your opponents with tact and respect. Always perform deliberate acts of kindness. Loyalty is never paid. Loyalty is being given out of deep respect for you as a person, of your vision and hard works. Never rely on loyalty that does not last long. You do not aspire to win for fame. Your goal is to win to help people. Ask respected higher ups for counsel periodically and spend time alone to think before you make any decision. A council is good but you are to pray for guidance from the most High One alone. God shall lead you as to how to perform your duties and how to help your people. Your basic strategy is to aim to do good, serve with compassion and do your works with passion. Make sure everyone knows you did the work by maintaining press releases, emails to higher ups and newsletters from your office. People have the right to be informed of how things are being run and you have every right to claim the good works that you have done. We are not to be silent with politics attacks on good works. But you are to be silent on politics attacks and smear campaigns on personal issues. We want to let you know that we can be silent with all the anticipated negative campaigns directed to us and to you. A man who damages the reputation of another man is a man that cannot be trusted. Remember my son to be careful not to do that in your political campaign no matter how tempting. Always be transparent and deal with the issues in an objective way. You are a man, be always a man. Friends come and go so choose well. Listen with your heart and not your ears. Choose the ones that will serve you with great dedication and loyalty. Real friendships are hard to find. Feel their honesty and seek for their commitment through good works. The most effective way to conquer the hearts of your people and conquer the respect of your opponents is by beating your opponents through good works. You are an open target to an open world. You are to protect yourself and your people as long as you can. You have your mission from God, you have work to do and we’ll always be here to wait for you to come home when you fail and feel like life seems to tumble down to keep you company. I will not ask any questions for any failure you may encounter. I am your mother who will always keep you in my heart and love you forever as my child. God speed my son. Do what you have to do for the love of God. Loving you always, Your mother How to cite Letter, Essay examples

World War I This short essay reflects the basic causes of WWI and the United States reasoning for entering WWI free essay sample

World War 1 was fought as a result of secret alliances formed in Europe in the late 19th and 20th centuries. The event which sparked the war was the assassination of Archduke Ferdinand on June 28th 1914 at the hands of Gabriel Principe. Archduke Ferdinand of Austria-Hungary was touring the nation of Serbia when a group of Bosnian nationalists named the Black Hand ordered his assassination. This act had a domino effect, and it quickly evolved into a war which spread rapidly through Europe. Secret alliances in Europe, an ever changing definition of neutrality in the United States, cultural issues, military considerations, and an explosive chain of events between the U. S. and Germany led American President Woodrow Wilson to ask Congress to declare war on Germany on April 6, 1917. In order to understand WW1, you must first understand the secret alliances among European nations. Austria-Hungary and Germany were allies. Germany issued â€Å"carte blanche† to Austria-Hungary which prompted Austria-Hungary to declare war on Serbia in retaliation to the assassination of Archduke Ferdinand. The Ottoman Empire entered the war on behalf of Austria-Hungary in order to gain territory from Russia and to gain control in the Balkans. Italy had a treaty with Germany, but later joined the allied powers in order to gain territory from Austria-Hungary. Serbia, which objected to Austria-Hungary’s demand to place troops within its borders, turned to Russia for support. When Russia mobilized its army, Germany demanded that it be demobilized. As a result of Russia’s refusal to do so, Germany attacked Russia. In reaction to Germany’s attack on Russia, France declared war on Germany because it had an alliance with Russia. In the interim, the Belgians were at war with Germany because Germany marched through Belgium without their permission in order to attack France. As a result, England (which had a treaty with Belgium,) declared war on Germany. The end result of this web of alliances was two opposing coalitions. The Triple Entente, also known as the Allied Powers, included Britain, France, Russia, and eventually Italy, Japan, and the U. S. The Triple Alliance, also known as the Central Powers, included Germany, Austria-Hungary, and The Ottoman Empire. Most Americans adopted a policy of Isolationism during WW1. They did not want wish to be involved in foreign affairs, least of all a war involving most of Europe. Pres. Wilson stated that the U. S. would remain neutral; however his definition of neutrality would change frequently during the course of the conflict. At the onset of the war, Wilson initiates a policy of complete neutrality. This means that not only would the U. S. refrain from assisting or trading with any of the warring nations, but it would ban private loans to them as well. By 1915, he realizes that this embargo could potentially plunge the U. S. into a recession, so he redefines neutrality as the ability to trade with all of the warring nations. This policy quickly goes sour because Britain forms a blockade to prevent the U. S. from trading with Germany. Germany counteracts the British blockade by using U-boats to attack ships carrying cargo to Britain. On May 7, 1915, a German U-boat fired a torpedo at the British passenger ship Lusitania killing 1198 passengers, 128 of which were Americans. The Germans claimed that this sinking was justifiable because there were materials on board. Pres. Wilson reacts by redefining neutrality as respecting the rights of neutral nations whether they decide to trade with everyone or no one. Germany initially complies with Pres. Wilsons wishes and pledges not to attack neutral ships. However, at this point in the conflict, the U. S. has ceased trade with Germany, and by 1916 is supplying the Allied Powers with 40% of their materiel in addition to loaning Britain and France money for the goods. In January 1917, Germany responds by adopting a policy of unconditional submarine warfare, targeting any ships it deemed a threat to its war effort. In the beginning of WW1, the U. S. policy of neutrality seemed the only viable option in order to keep the country out of the war while maintaining its trade relationships with the warring nations. However, the emergence of a world economy made it difficult for America’s economy to flourish without trading with Europe. With both the Allied and Central Powers in a stalemate, the only chance for either side to advance was to block trade with the U. S. Inevitably; the U. S. had to decide with whom it would trade, eventually choosing the Allied Powers. Once this decision was apparent, the Germans announced they would resume unrestricted submarine warfare. Subsequently, German Foreign Secretary Arthur Zimmerman sent a secret telegram to the German minister in Mexico, offering to help Mexico regain lost territory in the U. S. if Mexico declares war on the U. S. The â€Å"Zimmerman Telegram† was intercepted by the British on February 25, 1917 and the message was relayed to Pres. Wilson. Angered, Pres. Wilson asked Congress to approve an â€Å"armed neutrality† policy, allowing neutral ships to defend themselves against attackers. The following month, German U-boats sank 5 American ships off the coast of Britain, killing 66 Americans. Pres. Wilson felt the only option was to declare war on Germany, and did so on April 6, 1917 with the permission of Congress. Personally I believe WWI was extremely unnecessary and a war best described as carnage, it was crucial that the U. S. entered the war. If we had not then the allies would have most likely lost. The Germans out of line behavior particularly angered the American populace. Specifically the assassination of Edith Cavell a British nurse in Belgium while under German control. She and her Belgian accomplices were executed by The Germans at a rifle range for helping sneak wounded British soldiers back home. The Germans could have deported her back to the United Kingdom or kept her as a POW for her war crimes, but they chose to execute her instead. Even after the U. S. ambassador in Belgium at the time called German military officials and pleaded with them to spare their lives. It caused the Allied Powers to feel a great amount of disrespect. As well as the Zimmerman telegraph, which was Germany asking Mexico to give America trouble (Pancho Villa) and keep us out of their way, also left us feeling aggravated and disrespected towards Germany. And obviously the sinking of the Lusitania was by far the primary reason. Not a military ship at all, but a passenger ship simply returning passengers from the U. S. back to Britain was attacked and sunk by German U-boats not only killing innocent British passengers but innocent Americans as well. Bodies of dead passengers were washing up on the British coast for weeks. In my opinion I think the last straw for Pres. Wilson was his telegraph sent to Europe where he proposed to go to Europe personally and help end everything was denied. No winners no losers, just end all the nonsense now and start negotiating peace. It was turned down. After awhile all of these events compounded. We were being affected by the war before we were even in it, causing the majority of Americans to want to at that time. In fact we raised so much money selling liberty bonds to help fund the war the government had to ask the American people to stop buying and selling liberty bonds. There was more than enough support in entering and fighting in the war. We would have entered sooner, but Pres. Wilson put it off until he won his second election in 1916.

Working during high school

Introduction Many teens in high school get thrilled at the idea of working part time while still in school. The opportunity to make money is good and some parents give their children the liberty to work while still in school. For others, they would hear none of it because to them it is a complete waste of time. Whichever opinions parents may have the issue of students working throughout high school should be thought through carefully before making any decisions.Advertising We will write a custom essay sample on Working during high school specifically for you for only $16.05 $11/page Learn More Some students might find that their expectations about their prospective jobs are very different from the reality on the ground. However, if the students are open minded they will adjust well into the work place. Students should be allowed to work throughout high school to learn life skills and get a greater understanding of how the world operates once they are leg al adults. Networking Working while still in school presents students with a golden chance to start networking early in their life. The students who do part time jobs in companies get the opportunity to meet their future employers and if they become good employees while doing their stints as part time employees they will improve, their employability levels in the future because they will not be strangers in the same work places. Moreover, their former employers can offer them valuable assistance in case they need a reference from their current or former employers. The employers can also guide them in the right direction to take if they proof to be worthwhile employees. Students should also not shy away from working as volunteers in non profit organizations because such services might provide them with an opening in the future through making connections with people who might play a major role in their future careers. However, that should not be the primary goal of volunteering becaus e one should give expecting nothing in return. Learn the real world Students who work while still in high school get an opportunity to see how the real world operates and thus are better equipped than those who do not venture in the business world earlier. Working enables the students to integrate some of the things they learn theoretically in class in a real environment and makes them develop analytical skills as they deal with the challenges they encounter in their workplace. This increases their understanding, which may translate into better grades. Moreover, they gains skills that one cannot get in the classroom such as interaction with different clients or customers and thus they are equipped to deal with people from diverse backgrounds. Such students can comfortably work in any part of the world because their interactions with various people prepare them to become global citizens (Anderson Murphy, 6).Advertising Looking for essay on education? Let's see if we can help yo u! Get your first paper with 15% OFF Learn More Responsibility The opportunity to work makes students more responsible. A working student has to juggle three things that is education, work and play or leisure. Such a student must develop discipline to be able to balance the three things so that none suffers due to neglect because all are equally important. A student gets an opportunity to learn how to manage their time well and thus even in the future they will be able to handle their jobs well even if it means working extra time. Students also learn to be responsible with money because working teaches them the value of money unlike those who receive it from others and spend it without knowing its value or what it takes to earn. Make an extra coin Students get an opportunity to make some extra money and can buy stuff they admire which maybe their parents are unable to get them. Students come from diverse backgrounds in terms of economic resources. Some have adequate money a nd do not need to work at all. On the other hand, some come from disadvantaged backgrounds and must chip in to help the rest of the family and thus the opportunity to work while studying is valuable to them as it enables them to help their families or even cater for part of their school expenses. Academics may suffer Conversely, working throughout high school may affect a student’s academic performance adversely. Some students cannot cope with a work and class work and neglect one at the expense of the other. In such a scenario a student might fail in class work due to lack of concentration which might result from long working hours even though states regulate the number of hours students can work. The work schedule might conflict with the school timetable hence a students might be forced to skip some classes. Moreover, students may fail to engage in activities like sports, which require time in practice, and maybe they are good in a certain sport, which may earn them a study scholarship and give them an opportunity to further their education. Lack time for the family The working hours might eat into family time. Students may spend every free time they have including the weekends away on work and miss family activities. This is not healthy because people need to spend time together to build strong relationships. Parents who are busy at work most of the time do not get a chance to interact with their children too. Conclusion Finally, students should be allowed to work throughout high school because they get to learn a great deal about the real life. As long as a student can balance, their work and academics then they should be allowed to work. Working does students more good than harm and their parents should guide them in picking jobs that will help them in bettering their line of chosen careers. Will not only the students make an extra coin, but also learn to become responsible citizens and good members of their community in the future.Advertising We will write a custom essay sample on Working during high school specifically for you for only $16.05 $11/page Learn More Reference List Anderson, S. Murphy, N. (1999). Mandatory Community Service: Citizenship Education or Involuntary Servitude? Retrieved from ://www.ecs.org/html/Document.asp?chouseid=1426 This essay on Working during high school was written and submitted by user Mckenna Leach to help you with your own studies. You are free to use it for research and reference purposes in order to write your own paper; however, you must cite it accordingly. You can donate your paper here.

Et Al. Meaning and How to Use It

Et Al. Meaning and How to Use It Et al. essentially means â€Å"and others,† extra, or in addition. It is the abbreviated form of the Latin expression et alia (or et alii or et aliae, the masculine and feminine form of the plural, respectively). The abbreviation et al. often appears in academic documents. It is generally used in footnotes and citations: for example, when a book has multiple authors, et al. can be used after the first name to indicate that there are more than two other authors who worked on the project.   How to Use Et Al. Et al. can be used in a situation that refers to more than two people. Make sure it’s always followed with a period, which indicates that it’s an abbreviation, but given its prevalence in the English language, italicizing it is not necessary in reference citations, though some publications may require it. According to the APA, it should only be used when there are two or more authors. For three to five authors, all names must be listed within the first citation, but all following citations can include just the name of the first author and et al. For six or more authors, the first author and et al. can be used in all citations, including the first. If you’re referencing sources with many of the same authors, spell out as many names as possible before using et al., until there is no room for confusion. If using a different style guide, be sure to reference the corresponding manual as rules can differ. Keep in mind that since et al. is plural, it must apply to at least two people. For example, if you are dealing with four authors and have typed out three names, you cannot use et al. to substitute the last one, since it cannot be used in place of just one person. Does it have a place outside of citations? Generally, no. Though not technically incorrect, it would be rare, and overly formal, to see it within an email greeting to multiple people, such as: â€Å"Dear Bill et al.†Ã‚   Et Al. vs. Etc. Et al. might sound familiar to another abbreviation we encounter regularly: â€Å"etc.† Short for â€Å"et cetera†- which means â€Å"and the rest† in Latin- â€Å"etc.† refers to a list of things, rather than individuals. Unlike et al. which normally makes appearances in academic sources, â€Å"etc.† is both formal and informal and can be used in a wide variety of contexts. Examples of Et Al. Jolly et al. (2017) published a revolutionary study on the role of the gut microbiome: In this sentence, et al. doesn’t appear on a reference list, but still serves to indicate that Jolly and others contributed to the study in question.  Some large-scale surveys found cats to be the preferred pet (McCann et al., 1980) while others found dogs to be the ideal pet (Grisham Kane, 1981): In this example, et al. is used in the first citation because there are more than two authors. If this is a first citation, that indicates there are six or more authors, or if this is a subsequent citation in the text, there could be three or above authors. Et al. is not used in the last citation because there are only two authors who worked on the study.  Meditation once a week was found to improve focus by 20% in study participants (Hunter, Kennedy, Russell, Aarons, 2009). Meditation once a day was found to increase focus by 40% among participants (Hunter et al., 2009): This example, though citations of the same study would normally not occur in such close proximity, shows how et al. is used when introducing a work co-authored by three to five individuals. Et al. is reserved for all subsequent citations, with the first clearly naming everyone involved.   The Other â€Å"Et Al.†: Et Alibi In less common situations, et al. stands for et alibi, which refers to locations that will not appear in a list. For example, if you went on a trip, you could use et alibi when writing down the places and hotels you visited so you don’t have to name all of them. This can also be used to refer to locations within a text.   How do you remember what this means? Think of an alibi, which is used to prove that a criminal suspect was elsewhere when the crime took place, thus absolving them of suspicion.

Firewall Security Measures Essay Example | Topics and Well Written Essays - 3250 words

Firewall Security Measures - Essay Example A packet filter is the most simple type of firewall that operates at the network layer of OSI model. Packet filtering works on a set of rules stored as rule base, which determines which packet are allowed within the session, likewise which address are allowed for the communication process. If by default, a rule base does not permit any session, all packets are a drop from the communication. Information included in packet filtering are as follows: The source address of the packet (or the Layer 3 address) and the destination address of the packet (also Layer 3 address). Type of traffic or the specific network protocol (i.e. Ethernet) And possibly some information about the Layer 4 communication sessions (which is why packet filtering are sometimes considered to operate at layer 3 and 4 of OSI model). Stateful inspection, on the other hand, is the just superset of the packet filter. It also employs the method by which packet filtering works and an additional of storing the state of the session. For example, a session between 192.168.1.100:1023 and 210.9.88.29:80 was stored as â€Å"established† as its state, then the next time this session takes place, it will automatically be allowed. This provides a faster mechanism for filter incoming and outgoing session between server and host system. Stateful inspection firewall also operates at layer 3 and 4, plus layer 7 of the OSI model, which is evident on how stateful inspections consider application within the application layer.

Sustainable Engineering Assignment Essay Example | Topics and Well Written Essays - 250 words

Sustainable Engineering Assignment - Essay Example However, according to the table, natural gas produces 595 g of CO2 while generating 1kWh of energy. It depicts the initial value to be high and without heat recovery method. b) According to the table hard coal produces about 29 mg of Dinitrogen oxides, 1.5 g methane and 1144 grams of CO2, while generating 1kWh of electrical energy. On the other hand, natural gas produces about 12 mg of Dinitrogen oxides, 3.4 g methane and 595 grams of CO2, while generating 1kWh of electrical energy. Hard coal produces much larger amount of emissions as compared to the natural gas. Thus, natural gas is environmentally friendlier as compared to the hard coal. c) If the amount of leakage exceeds with a fraction of 3.7, the emissions of the shale gas to generate 1kWh of electrical energy, equals to the emissions from the hard coal to generate 1kWh of electrical energy. The fraction will increase the methane emission of the shale gas to 551

Theorems Related To Mersenne Primes Mathematics Essay

Theorems Related To Mersenne Primes Mathematics Essay Introduction: In the past many use to consider that the numbers of the type 2p-1 were prime for all primes numbers which is p, but when Hudalricus Regius (1536) clearly established that 211-1 = 2047 was not prime because it was divisible by 23 and 83 and later on Pietro Cataldi (1603) had properly confirmed about 217-1 and 219-1 as both give prime numbers but also inaccurately declared that 2p-1 for 23, 29, 31 and 37 gave prime numbers. Then Fermat (1640) proved Cataldi was wrong about 23 and 37 and Euler (1738) showed Cataldi was also incorrect regarding 29 but made an accurate conjecture about 31. Then after this extensive history of this dilemma with no accurate result we saw the entry of Martin Mersenne who declared in the introduction of his Cogitata Physica-Mathematica (1644) that the numbers 2p-1 were prime for:- p= 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257 and for  other positive integers where p So simply the definition is when 2p-1 forms a prime number it is recognized to be a Mersenne prime. Many years later with new numbers being discovered belonging to Mersenne Primes there are still many fundamental questions about Mersenne primes which remain unresolved. It is still not identified whether Mersenne primes is infinite or finite. There are still many aspects, functions it performs and applications of Mersenne primes that are still unfamiliar With this concept in mind the focus of my extended essay would be: What are Mersenne Primes and it related functions? The reason I choose this topic was because while researching on my extended essay topics and I came across this part which from the beginning intrigued me and it gave me the opportunity to fill this gap as very little was taught about these aspects in our school and at the same time my enthusiasm to learn something new through research on this topic. Through this paper I will explain what are Mersenne primes and certain theorems, related to other aspects and its application that are related with it. Theorems Related to Mersenne Primes: p is prime only if 2p  Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢Ã‚  1 is prime. Proof: If p is composite then it can be written as p=x*y with x, y > 1. 2xy-1= (2x-1)*(1+2x+22x+23x+à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦..+2(b-1)a) Thus we have got 2xy à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 as a product of integers > 1. If n is an odd prime, then any prime m that divides 2n à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 must be 1 plus a multiple of 2n. This holds even when 2n à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 is prime. Examples: Example I: 25 à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 = 31 is prime, and 31 is multiple of (2ÃÆ'-5) +1 Example II: 211 à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 = 23ÃÆ'-89, where 23 = 1 + 2ÃÆ'-11, and 89 = 1 + 8ÃÆ'-11. Proof: If m divides 2n à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 then 2n à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod m). By Fermats Theorem we know that 2(m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod m). Assume n and m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 are comparatively prime which is similar to Fermats Theorem that states that (m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1)(n à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod n). Hence there is a number x à ¢Ã¢â‚¬ °Ã‚ ¡ (m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1)(n à ¢Ã‹â€ Ã¢â‚¬â„¢ 2) for which (m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) ·x à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod n), and thus a number k for which (m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) ·x à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 = kn. Since 2(m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod m), raising both sides of the congruence to the power x gives 2(m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1)x à ¢Ã¢â‚¬ °Ã‚ ¡ 1, and since 2n à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod m), raising both sides of the congruence to the power k gives 2kn à ¢Ã¢â‚¬ °Ã‚ ¡ 1. Thus 2(m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1)x/2kn = 2(m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1)x à ¢Ã‹â€ Ã¢â‚¬â„¢ kn à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod m). But by meaning, ( m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1)x à ¢Ã‹â€ Ã¢â‚¬â„¢ kn = 1 which implies that 21 à ¢Ã¢â‚¬ °Ã‚ ¡ 1 (mod m) which means that m divides 1. Thus the first conjecture that n and m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 are relatively prime is unsustainable. Since n is prime m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 have to be a multiple of n. Note: This information provides a confirmation of the infinitude of primes different from Euclids Theorem which states that if there were finitely many primes, with n being the largest, we have a contradiction because every prime dividing 2n à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 must be larger than n. If n is an odd prime, then any prime m that divides 2n à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 must be congruent to +/-1 (mod 8). Proof: 2n + 1 = 2(mod m), so 2(n + 1) / 2 is a square root of 2 modulo m. By quadratic reciprocity, any prime modulo which 2 has a square root is congruent to +/-1 (mod 8). A Mersenne prime cannot be a Wieferich prime. Proof: We show if p = 2m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 is a Mersenne prime, then the congruence does not satisfy. By Fermats Little theorem, m | p à ¢Ã‹â€ Ã¢â‚¬â„¢ 1. Now write, p à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 = mÃŽÂ ». If the given congruence satisfies, then p2 | 2mÃŽÂ » à ¢Ã‹â€ Ã¢â‚¬â„¢ 1, therefore Hence 2m à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 | ÃŽÂ », and therefore . This leads to , which is impossible since . The Lucas-Lehmer Test Mersenne prime are found using the following theorem: For n an odd prime, the Mersenne number 2n-1 is a prime if and only if 2n -1 divides S(p-1) where S(p+1) = S(p)2-2, and S(1) = 4. The assumption for this test was initiated by Lucas (1870) and then made into this straightforward experiment by Lehmer (1930). The progression S(n) is calculated modulo 2n-1 to conserve time.   This test is perfect for binary computers since the division by 2n-1 (in binary) can only be completed using rotation and addition. Lists of Known Mersenne Primes: After the discovery of the first few Mersenne Primes it took more than two centuries with rigorous verification to obtain 47 Mersenne primes. The following table below lists all recognized Mersenne primes:- It is not well-known whether any undiscovered Mersenne primes present between the 39th and the 47th from the above table; the position is consequently temporary as these numbers werent always discovered in their increasing order. The following graph shows the number of digits of the largest known Mersenne primes year wise. Note: The vertical scale is logarithmic. Factorization The factorization of a prime number is by meaning itself the prime number itself. Now if talk about composite numbers. Mersenne numbers are excellent investigation cases for the particular number field sieve algorithm, so frequently that the largest figure they have factorized with this has been a Mersenne number. 21039 1 (2007) is the record-holder after estimating took with the help of a couple of hundred computers, mostly at NTT in Japan and at EPFL in Switzerland and yet the time period for calculation was about a year. The special number field sieve can factorize figures with more than one large factor. If a number has one huge factor then other algorithms can factorize larger figures by initially finding the answer of small factors and after that making a primality test on the cofactor. In 2008 the largest Mersenne number with confirmed prime factors is 217029 à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 = 418879343 ÃÆ'- p, where p was prime which was confirmed with ECPP. The largest with possible pr ime factors allowed is 2684127 à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 = 23765203727 ÃÆ'- q, where q is a likely prime. Generalization: The binary depiction of 2p à ¢Ã‹â€ Ã¢â‚¬â„¢ 1 is the digit 1 repeated p times. A Mersenne prime is the base 2 repunit primes. The base 2 depiction of a Mersenne number demonstrates the factorization example for composite exponent. Examples in binary notation of the Mersenne prime would be: 25à ¢Ã‹â€ Ã¢â‚¬â„¢1 = 111112 235à ¢Ã‹â€ Ã¢â‚¬â„¢1 = (111111111111111111111111111111)2 Mersenne Primes and Perfect Numbers Many were anxious with the relationship of a two sets of different numbers as two how they can be interconnected. One such connection that many people are concerned still today is Mersenne primes and Perfect Numbers. When a positive integer that is the sum of its proper positive divisors, that is, the sum of the positive divisors excluding the number itself then is it said to be known as Perfect Numbers. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors. There are said to be two types of perfect numbers: 1) Even perfect numbers- Euclid revealed that the first four perfect numbers are generated by the formula 2nà ¢Ã‹â€ Ã¢â‚¬â„¢1(2n  Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢Ã‚  1): n = 2:    2(4 à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) = 6 n = 3:    4(8 à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) = 28 n = 5:    16(32 à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) = 496 n = 7:    64(128 à ¢Ã‹â€ Ã¢â‚¬â„¢ 1) = 8128. Noticing that 2n  Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢Ã‚  1 is a prime number in each instance, Euclid proved that the formula 2nà ¢Ã‹â€ Ã¢â‚¬â„¢1(2n  Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢Ã‚  1) gives an even perfect number whenever 2p  Ãƒ ¢Ã‹â€ Ã¢â‚¬â„¢Ã‚  1 is prime 2) Odd perfect numbers- It is unidentified if there might be any odd perfect numbers. Various results have been obtained, but none that has helped to locate one or otherwise resolve the question of their existence. An example would be the first perfect number that is 6. The reason for this is so since 1, 2, and 3 are its proper positive divisors, and 1  +  2  +  3  =  6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: (1  +  2  +  3  +  6)  /  2  =  6. Few Theorems related with Perfect numbers and Mersenne primes: Theorem One: z is an even perfect number if and only if it has the form 2n-1(2n-1) and 2n-1 is a prime. Suppose first that   p = 2n-1 is a prime number, and set l = 2n-1(2n-1).   To show l is perfect we need only show sigma(l) = 2l.   Since sigma is multiplicative and sigma(p) = p+1 = 2n, we know sigma(n) = sigma(2n-1).sigma(p) =  (2n-1)2n = 2l. This shows that l is a perfect number. On the other hand, suppose l is any even perfect number and write l as 2n-1m where m is an odd integer and n>2.   Again sigma is multiplicative so sigma(2n-1m) = sigma(2n-1).sigma(m) = (2n-1).sigma(m). Since l is perfect we also know that sigma(l) = 2l = 2nm. Together these two criteria give 2nm = (2n-1).sigma(m), so 2n-1 divides 2nm hence 2n-1 divides m, say m = (2n-1)M.   Now substitute this back into the equation above and divide by 2n-1 to get 2nM = sigma(m).   Since m and M are both divisors of m we know that 2nM = sigma(m) > m + M = 2nM, so sigma(m) = m + M.   This means that m is prime and its only two divisors are itself (m) and one (M).   Thus m = 2n-1 is a prime and we have prove that the number l has the prescribed form. Theorem Two: n will also be a prime if 2n-1 is a prime. Proof: Let r and s be positive integers, then the polynomial xrs-1 is xs-1 times xs(r-1) + xs(r-2) + + xs + 1.   So if n is composite (say r.s with 1 Theorem Three:   Let n and m be primes. If q divides Mn = 2n-1, then q = +/-1 (mod 8)  Ã‚  Ã‚  Ã‚   and  q = 2kn + 1 for some integer k. Proof: If p divides Mq, then 2q  =  1 (mod p) and the order of 2 (mod p) divides the prime q, so it must be q.   By Fermats Little Theorem the order of 2 also divides p-1, so p-1  =  2kq.   This gives 2(p-1)/2 = 2qk = 1 (mod p) so 2 is a quadratic residue mod p and it follows p = +/-1 (mod 8), which completes the proof. Theorem Four: If p = 3 (mod 4) be prime and then 2p+1 is also prime only if 2p+1 divides 2p-1. Proof: Suppose q = 2p+1 is prime. q  =  7 (mod  8) so 2 is a quadratic residue modulo q and it follows that there is an integer n such that n2  =  2 (mod  q). This shows 2p = 2(q-1)/2 = nq-1 = 1 (mod q), showing q divides Mp.       Conversely, let 2p+1 be a factor of Mp. Suppose, for proof by contradiction, that 2p+1 is composite and let q be its least prime factor. Then 2p  =  1 (mod  q) and the order of 2 modulo q divides both p and q-1, hence p divides q-1. This shows q  >  p and it follows (2p+1) + 1 > q2 > p2 which is a contradiction since p > 2. Theorem Five: When we add the digits of any even perfect number with the exception of 6 and then sum the digits of the resulting number and keep doing it again until we get a single digit which will be one. Examples. 28  ¬10  ¬ 1, 496  ¬ 19  ¬ 10  ¬ 1, and 8128  ¬ 19  ¬10  ¬ 1 Proof: Let s(n) be the sum of the digits of n. It is easy to see that s(n) = n (mod 9). So to prove the theorem, we need only show that perfect numbers are congruent to one modulo nine. If n is a perfect number, then n has the form 2p-1(2p-1) where p is prime which see in the above theorem one. So p is either 2, 3, or is congruent to 1 or 5 modulo 6. Note that we have excluded the case p=2 (n=6). Finally, modulo nine, the powers of 2 repeat with period 6 (that is, 26 = 1 (mod 9)), so modulo nine n is congruent to one of the three numbers 21-1(21-1), 23-1(23-1), or 25-1(25-1), which are all 1 (mod 9). Conjectures and Unsolved Problems: Does an odd perfect number exist?   We have so far known that even perfect numbers are 2n-1(2n-1)from the Theorem One above, but what about odd perfect numbers?   If there is an odd perfect number, then it has to follow certain conditions:- To be a perfect square times an odd power of a single prime; It is divisible by at least eight primes and has to have at least 75 prime factors with at least 9 distinct It has at least 300 decimal digits and it has a prime divisor greater that 1020. Are there infinite numbers of Mersenne primes?   The answer is probably yes because of the harmonic sequence deviation. The New Mersenne Conjecture: P. T. Bateman, J. L. Selfridge and Wagstaff, Jr., S. S., have conjectured the following:- Let n be any odd natural number. If two of the following statements hold, subsequently so does the third: n = 2p+/-1  Ã‚   or  Ã‚   n = 4p+/-3 2n-1 is a prime (2n+1)/3 is a prime. Are all Mersenne number 2n-1 square free? This is kind of like an open question to which the answer is still not known and hence it cannot be called a conjecture. It is simple to illustrate that if the square of a prime n divides a Mersenne, then p is a Wieferich prime which are uncommon!   Only two are acknowledged lower than 4,000,000,000,000 and none of these squared divide a Mersenne.    If C0 = 2, then let C1 = 2C0-1, C2 = 2C1-1, C3 = 2C2-1à ¢Ã¢â€š ¬Ã‚ ¦Ãƒ ¢Ã¢â€š ¬Ã‚ ¦ then are all of these prime numbers?   Dickson Catalan (1876) responded to Lucas stating 2127-1 (which is C4) being a prime with this sequence: C0 = 2 (which is a prime) C1 = 3 (which is a prime) C2 = 7 (which is a prime) C3 = 127 (which is a prime) C4 = 170141183460469231731687303715884105727 (which is a prime) C5 > 1051217599719369681875006054625051616349 (is C5 a prime or not?) It looks as if it will not be very likely that C5 or further larger terms would be prime number.   If there is a single composite term in this series, then by theorem one each and every one of the following terms would be composite.   Are there more double-Mersenne primes? Another general misunderstanding was that if n=Mp is prime, then so is Mn; Lets assume this number Mn to be MMp which would be a double-Mersenne.  As we apply this to the first four such numbers we get prime numbers: MM2 = 2(4  -1) -1= 23-1  Ã‚   =  7 MM3 =  2(8-1)-1  Ã‚   =  127 MM5 =  2(32-1)-1  =  2147483647, MM7 =  2(128-1)-1 =  170141183460469231731687303715884105727. Application of Mersenne Prime: In computer science, unspecified p-bit integers can be utilized to express numbers up to Mp. In the mathematical problem Tower of Hanoi is where the Mersenne primes are used. It is a mathematical puzzle consisting of three rods, and a number of disks of different sizes, which can slide onto any rod. The puzzle begins with the disks in ascending order of size on the first rod, the largest at the bottom to the smallest at the top. A diagram given below illustrates the Tower of Hanoi. The objective of the puzzle is to move the entire stack to another rod, obeying the following rules: Only one disk may be moved at a time. Each move consists of taking the upper disk from one of the rods and sliding it onto another rod, on top of the other disks that may already be present on that rod. No disk may be placed on top of a smaller disk. Now to solve this game with a p-disc tower needs the minimum of Mp no of steps, where p is the no of disc used in the Tower of Hanoi and if we use the formula of Mersenne then we get the required result. An example of this would be if there were 5 discs involved in this Tower of Hanoi then the least number of steps required to finish this game would be 31 steps minimum. Conclusion After investigating the entire aspects, functions, and few applications of Mersenne Primes I believe that there is still many unsolved theories when it comes to Mersenne primes. These primes are also useful to investigates much further and deeper into the number system and help us to understand more sets of numbers such as Fermat prime, Wieferich prime, Wagstaff prime, Solinas prime etc.