Health Promotion Assignment Example | Topics and Well Written Essays - 750 words - 1

Health Promotion - Assignment Example The paper was supposed to begin by introducing the whole content of the paper in an abstract that is subdivided into subheadings representing the major heading of the report. This, however, was adhered to. The authors reported this in a manner that captivates the attention of the reader by letting the audience know what the paper entails and making the decision on whether to proceed with reading the whole paper or not depending on his interest. The abstract had the purpose, participants, method, result, conclusion and key words meeting the desired reporting guideline for the same. From this section a lone the researcher made known the reason he was engaging in the research reason being to explore the patients' perspective regarding their decision to seek health care services or avoid them altogether(Carla, Green, Johnson, & Yarborough, 2014).In the introduction section, more of what was covered in the abstract were expounded with more explanation about the need for that study. The re search design also covered how the survey was conducted by from October 2002 to April 2003 with 150 participants. The researchers reported having assessed that the participants had 12 months old plan. The mean of the population interviewed was also reported and the consideration given to the participants for their time was also acknowledged. This supports the validity of the response given since participants will not be worried about the time they are losing out from doing the economic activity.

Final Global Media writing task Final Exam 500 words Essay

Final Global Media writing task Final Exam 500 words - Essay Example I learnt that digital media for instance the internet, has played a major role in simplification of worldwide transmission of information. Internet is easily accessible in most parts of the world and therefore provides a platform for easy transmission of information in numerous parts of the world instantaneously. In the course Global Media Trends, I had the chance of acquainting myself with various issues pertaining global media presently. For instance, I realized that despite solving the problem of information transmission, global media continues to face challenges. The project on, â€Å"The growth in the concentration of media ownership around the world: A Case Study of Rupert Murdoch’s News Corp†, presented me with a chance to realize one of the current problems facing global media. The problem of media monopoly in the ownership of media leads to unethical practices. Media in a very critical part of the society and any unethical practices would lead to sabotage of societal rights. Through this course, I learned various ways in which developments in global media have simplified ways of sharing information throughout the world. The group assignment about online activism in particular widened my knowledge on how improvements in technology have made a great contribution in global media. Through online activism, it is possible to pass various messages to many people throughout the world. For instance, one is able to create awareness regarding an issue like pollution by publishing information online. The YouTube assignment made me realize how advancements in global media have contributed improvement of the education sector. Through YouTube, students can share education videos with their colleagues. Others get the chance of accessing scholarly videos at no cost. This course made me realize that, advancements in the global media, apart from

Philosophy of Nursing Essay Example | Topics and Well Written Essays - 750 words - 1

Philosophy of Nursing - Essay Example The life philosophy may vary based on individual differences, cultural differences and the religious differences. Philosophy is a kind of beliefs, values and goals a person sets in his life. Philosophy in short is the personal belief about how to live or how to deal with a situation. On academic terms, philosophy is the origin of every sort of knowledge. In olden times, when science was not developed much, the philosophers were the persons who interpreted the problems and they were able to provide answers to the doubts of the common people. Philosophy was the first sort of knowledge originated in human history. All the other subjects like, science, mathematics, literature etc gradually separated from philosophy when the knowledge became gradually increased. Since the origin of every sort of knowledge is attributed to philosophy, the highest degree awarded in all the subjects is labelled as Doctor of Philosophy or PhD in academics. Two persons working in an organization, in the same department and same position may approach their work differently because of their difference in attitudes and philosophy about work. For example, consider two accountants working in the same company. One may finish his works daily and may not keep anything for the next day. The other may not finish his works in time and he may keep something for the next day. This is because of their difference in attitude or philosophy in approaching the profession. Nursing profession is considered as a noble profession because of their contributions in saving the life of sick people. For me nurses are not getting the respect they deserve from the society considering the nature of their services. I have selected nursing as my profession because of its relevance and value in the society. For me human life is the most important thing in the world and I am happy, if I am

Recyclable Drinking Water Essay Example | Topics and Well Written Essays - 3250 words

Recyclable Drinking Water - Essay Example The city chosen for this purpose is Bangkok considering the current flood scenario that has affected the lives of locals and tourists. The shortage of drinking water has jeopardised the physical and mental state along with the health of people. Bangkok is the capital and largest city of Thailand known as a tourist destination across the globe. The influx of tourists from different parts of the world has strengthened the economy of the country boasting on impeccable infrastructure and hospitality that is good enough to allure people across the globe. Bangkok has 50 districts and each district is managed by the district chief appointed by the governor. There is also an elected government body; Bangkok Metropolitan Council that managed municipal ordinances and city’s budget pertaining to infrastructure and other activities. The government of Bangkok of is known as Bangkok Metropolitan Administration that manages the city and its resources in an effective and efficient manner. The re are a number of national and international retailers present in Bangkok with the likes of Tesco, Family Mart and Big C dominating the market. The economic activities are majorly restricted to agriculture and tourism. In Bangkok, Metropolitan Waterworks Authority offers tap water that is used for the purpose of drinking and other activities. Many households have water purifying machines that further purifies water through different mechanisms and processes. At the same time, a number of packaged drinking water bottles are available that can be bought to ensure good quality of drinking water. However, there is no organisation developing or promoting recycled water for the drinking purpose. This marketing plan would ensure successful introduction of the product in the markets of Bangkok... The paper tells about integrated marketing communication as one of the highly preferred marketing tools within the competitive business environment. It can be defined as the coordination and amalgamation of marketing communication tools and functions into a program or function that helps in impacting customers along with enhancing the feasibility and acceptance of the product. However, the common perception states that recycle water is not safe for drinking and can be used in irrigation, industrial usage and flushing toilets. Bangkok has been severely affected by floods creating shortages for safe drinking water. It needs to be mentioned that common people in the city are facing serious water crisis with tap water pouring contaminated and dirty water. On the other hand, packaged drinking bottles are quite limited in the market and the city does not have a marketer offering recycled drinking water. It can be assumed that development of recycled drinking water would help in offering a useful product to the masses using an integrated marketing communication approach. This would also help in assessing the present use of recycled water and its importance and acceptance in the business and social environment. Thus, the marketing plan has a great scope that requires effective planning and implementation. The city of Bangkok has been developing wastewater treatment system from a long period of time and efforts have yielded good benefits in last few years. There are a number of sewerage zones that have helped in improving the overall water shortages in the short as well as in the long run.

Film industry in china Essay Example | Topics and Well Written Essays - 1250 words

Film industry in china - Essay Example Practically, issues concerning films began back in 1896 when the natives started producing motion pictures in a place called Shanghai. This led to the production of the first film in 1905 name the Beijing Opera: The Battle of Dingjunshan. They were several films produced following the release of the first film but foreigners owned them. This was until 1916 when a Chinese native in Shanghai set up the first production company. As the industries grew, American film producers visited the Shanghai, which was the center for film production. Americans were much ahead in the industry and thus acted a clear guide to the growing industry. This led to the setup of a training center at Shanghai in 1920s (Curtin 45). Under the American patronage, China was able to produce its first true film in 1930s. There were a series of films produced during this period following the increases level of experience and human resource to boost the industry. In late 1940s and early 1950s, the industry grew follo wing the establishment of a substantial Chinese production house named Lianhau Company. There was also an increase the number of movie viewers as they increased from 140 million to 4.5 billion (Nakajima 23). Chinese is currently one of the leading nations in the world in producing films. Practically, Chinese are diligent, and they have taken the industry to international levels despite current hardened censorship placed by the centralized government. Key players in the film industry are using illegal means to sell their products globally especially in USA and Europe. This has popularized their expertise in the field, thus out doing American who was previously thought to have the largest film industry in the world. To enhance globalization of the industry, they have also adopted a trend where they produce films in foreign nations, as this will bar the government from interfering with the proceeding in the industry. China’s film and movie industry is

Air Pollution Essay Example for Free

Air Pollution Essay Air Pollution, addition of harmful substances to the atmosphere resulting in damage to the environment, human health, and quality of life. One of many forms of pollution, air pollution occurs inside homes, schools, and offices; in cities; across continents; and even globally. Air pollution makes people sick—it causes breathing problems and promotes cancer—and it harms plants, animals, and the ecosystems in which they live. Some air pollutants return to Earth in the form of acid rain and snow, which corrode statues and buildings, damage crops and forests, and make lakes and streams unsuitable for fish and other plant and animal life. Pollution is changing Earth’s atmosphere so that it lets in more harmful radiation from the Sun. At the same time, our polluted atmosphere is becoming a better insulator, preventing heat from escaping back into space and leading to a rise in global average temperatures. Scientists predict that the temperature increase, referred to as global warming, will affect world food supply, alter sea level, make weather more extreme, and increase the spread of tropical disease. Most air pollution comes from one human activity: burning fossil fuels—natural gas, coal, and oil—to power industrial processes and motor vehicles. Among the harmful chemical compounds this burning puts into the atmosphere are carbon dioxide, carbon monoxide, nitrogen oxides, sulfur dioxide, and tiny solid particles—including lead from gasoline additives—called particulates. Between 1900 and 1970, motor vehicle use rapidly expanded, and emissions of nitrogen oxides, some of the most damaging pollutants in vehicle exhaust, increased 690 percent. When fuels are incompletely burned, various chemicals called volatile organic chemicals (VOCs) also enter the air. Pollutants also come from other sources. For instance, decomposing garbage in landfills and solid waste disposal sites emits methane gas, and many household products give off VOCs. Some of these pollutants also come from natural sources. For example, forest fires emit particulates and VOCs into the atmosphere. Ultrafine dust particles, dislodged by soil erosion when water and weather loosen layers of soil, increase airborne particulate levels. Volcanoes spew out sulfur dioxide and large amounts of pulverized lava rock known as volcanic ash. A big volcanic eruption can darken the sky over a wide region and affect the Earth’s entire atmosphere. The 1991 eruption of Mount Pinatubo in the Philippines, for example, dumped enough volcanic ash into the upper atmosphere to lower global temperatures for the next two years. Unlike pollutants from human activity, however, naturally occurring pollutants tend to remain in the atmosphere for a short time and do not lead to permanent atmospheric change. Once in the atmosphere, pollutants often undergo chemical reactions that produce additional harmful compounds. Air pollution is subject to weather patterns that can trap it in valleys or blow it across the globe to damage pristine environments far from the original sources. Local and regional pollution take place in the lowest layer of the atmosphere, the troposphere, which at its widest extends from Earths surface to about 16 km (about 10 mi). The troposphere is the region in which most weather occurs. If the load of pollutants added to the troposphere were equally distributed, the pollutants would be spread over vast areas and the air pollution might almost escape our notice. Pollution sources tend to be concentrated, however, especially in cities. In the weather phenomenon known as thermal inversion, a layer of cooler air is trapped near the ground by a layer of warmer air above. When this occurs, normal air mixing almost ceases and pollutants are trapped in the lower layer. Local topography, or the shape of the land, can worsen this effect—an area ringed by mountains, for example, can become a pollution trap. Smog is intense local pollution usually trapped by a thermal inversion. Before the age of the automobile, most smog came from burning coal. In 19th-century London, smog was so severe that street lights were turned on by noon because soot and smog darkened the midday sky. Burning gasoline in motor vehicles is the main source of smog in most regions today. Powered by sunlight, oxides of nitrogen and volatile organic compounds react in the atmosphere to produce photochemical smog. Smog contains ozone, a form of oxygen gas made up of molecules with three oxygen atoms rather than the normal two. Ozone in the lower atmosphere is a poison—it damages vegetation, kills trees, irritates lung tissues, and attacks rubber. Environmental officials measure ozone to determine the severity of smog. When the ozone level is high, other pollutants, including carbon monoxide, are usually present at high levels as well (see Air Quality). In the presence of atmospheric moisture, sulfur dioxide and oxides of nitrogen turn into droplets of pure acid floating in smog. These airborne acids are bad for the lungs and attack anything made of limestone, marble, or metal. In cities around the world, smog acids are eroding precious artifacts, including the Parthenon temple in Athens, Greece, and the Taj Mahal in Agra, India. Oxides of nitrogen and sulfur dioxide pollute places far from the points where they are released into the air. Carried by winds in the troposphere, they can reach distant regions where they descend in acid form, usually as rain or snow. Such acid precipitation can burn the leaves of plants and make lakes too acidic to support fish and other living things. Because of acidification, sensitive species such as the popular brook trout can no longer survive in many lakes and streams in the eastern United States. Smog spoils views and makes outdoor activity unpleasant. For the very young, the very old, and people who suffer from asthma or heart disease, the effects of smog are even worse: It may cause headaches or dizziness and can cause breathing difficulties. In extreme cases, smog can lead to mass illness and death, mainly from carbon monoxide poisoning. In 1948 in the steel-mill town of Donora, Pennsylvania, intense local smog killed 19 people. In 1952 in London about 4,000 people died in one of the notorious smog events known as London Fogs; in 1962 another 700 Londoners died. With stronger pollution controls and less reliance on coal for heat, today’s chronic smog is rarely so obviously deadly. However, under adverse weather conditions, accidental releases of toxic substances can be equally disastrous. The worst such accident occurred in 1984 in Bhopal, India, when methyl isocyanate released from an American-owned factory during a thermal inversion caused more than 3,800 deaths. Air pollution can expand beyond a regional area to cause global effects. The stratosphere is the layer of the atmosphere between 16 km (10 mi) and 50 km (30 mi) above sea level. It is rich in ozone, the same molecule that acts as a pollutant when found at lower levels of the atmosphere in urban smog. Up at the stratospheric level, however, ozone forms a protective layer that serves a vital function: It absorbs the wavelength of solar radiation known as ultraviolet-B (UV-B). UV-B damages deoxyribonucleic acid (DNA), the genetic molecule found in every living cell, increasing the risk of such problems as cancer in humans. Because of its protective function, the ozone layer is essential to life on Earth. Several pollutants attack the ozone layer. Chief among them is the class of chemicals known as chlorofluorocarbons (CFCs), formerly used as refrigerants (notably in air conditioners), as agents in several manufacturing processes, and as propellants in spray cans. CFC molecules are virtually indestructible until they reach the stratosphere. Here, intense ultraviolet radiation breaks the CFC molecules apart, releasing the chlorine atoms they contain. These chlorine atoms begin reacting with ozone, breaking it down into ordinary oxygen molecules that do not absorb UV-B. The chlorine acts as a catalyst—that is, it takes part in several chemical reactions—yet at the end emerges unchanged and able to react again. A single chlorine atom can destroy up to 100,000 ozone molecules in the stratosphere. Other pollutants, including nitrous oxide from fertilizers and the pesticide methyl bromide, also attack atmospheric ozone. Scientists are finding that under this assault the protective ozone layer in the stratosphere is thinning. In the Antarctic region, it vanishes almost entirely for a few weeks every year. Although CFC use has been greatly reduced in recent years and will soon be prohibited worldwide, CFC molecules already released into the lower atmosphere will be making their way to the stratosphere for decades, and further ozone loss is expected. As a result, experts anticipate an increase in skin cancers, more cataracts (clouding of the lens of the eye), and reduced yields of some food crops. Humans are bringing about another global-scale change in the atmosphere: the increase in what are called greenhouse gases. Like glass in a greenhouse, these gases admit the Sun’s light but tend to reflect back downward the heat that is radiated from the ground below, trapping heat in the Earth’s atmosphere. This process is known as the greenhouse effect. Carbon dioxide is the most significant of these gases—there is 31 percent more carbon dioxide in the atmosphere today than there was in 1750, the result of our burning coal and fuels derived from oil. Methane, nitrous oxide, and CFCs are greenhouse gases as well. Scientists predict that increases in these gases in the atmosphere will make the Earth a warmer place. They expect a global rise in average temperature of 1. 4 to 5. 8 Celsius degrees (2. 5 to 10. 4 Fahrenheit degrees) in the next century. Average temperatures have in fact been rising, and the 1990s were the warmest decade on record. Some scientists are reluctant to say that global warming has actually begun because climate naturally varies from year to year and decade to decade, and it takes many years of records to be sure of a fundamental change. There is little disagreement, though, that global warming is on its way. Global warming will have different effects in ifferent regions. A warmed world is expected to have more extreme weather, with more rain during wet periods, longer droughts, and more powerful storms. Although the effects of future climate change are unknown, some predict that exaggerated weather conditions may translate into better agricultural yields in areas such as the western United States, where temperature and rainfall are expected to increase, while dramatic decreases in rainfall may lead to severe drought and plunging agricultural yields in parts of Africa, for example. Warmer temperatures are expected to partially melt the polar ice caps, leading to a projected sea level rise of 9 to 100 cm (4 to 40 in) by the year 2100. A sea level rise at the upper end of this range would flood coastal cities, force people to abandon low-lying islands, and completely inundate coastal wetlands. If sea levels rise at projected rates, the Florida Everglades could be completely under salt water in the next century. Diseases like malaria, which at present are primarily found in the tropics, may become more common in the regions of the globe between the tropics and the polar regions, called the temperate zones. For many of the world’s plant species, and for animal species that are not easily able to shift their territories as their habitat grows warmer, climate change may bring extinction. Pollution is perhaps most harmful at an often unrecognized site—inside the homes and buildings where we spend most of our time. Indoor pollutants include tobacco smoke; radon, an invisible radioactive gas that enters homes from the ground in some regions; and chemicals released from synthetic carpets and furniture, pesticides, and household cleaners. When disturbed, asbestos, a nonflammable material once commonly used in insulation, sheds airborne fibers that can produce a lung disease called asbestosis. Pollutants may accumulate to reach much higher levels than they do outside, where natural air currents disperse them. Indoor air levels of many pollutants may be 2 to 5 times, and occasionally more than 100 times, higher than outdoor levels. These levels of indoor air pollutants are especially harmful because people spend as much as 90 percent of their time living, working, and playing indoors. Inefficient or improperly vented heaters are particularly dangerous. In the United States, the serious effort against local and regional air pollution began with the Clean Air Act of 1970, which was amended in 1977 and 1990. This law requires that the air contain no more than specified levels of particulate matter, lead, carbon monoxide, sulfur dioxide, nitrogen oxides, volatile organic compounds, ozone, and various toxic substances. To avoid the mere shifting of pollution from dirty areas to clean ones, stricter standards apply where the air is comparatively clean. In national parks, for instance, the air is supposed to remain as clean as it was when the law was passed. The act sets deadlines by which standards must be met. The Environmental Protection Agency (EPA) is in charge of refining and enforcing these standards, but the day-to-day work of fighting pollution falls to the state governments and to local air pollution control districts. Some states, notably California, have imposed tougher air pollution standards of their own. In an effort to enforce pollution standards, pollution control authorities measure both the amounts of pollutants present in the atmosphere and the amounts entering it from certain sources. The usual approach is to sample the open, or ambient, air and test it for the presence of specified pollutants. The amount of each pollutant is counted in parts per million or, in some cases, milligrams or micrograms per cubic meter. To learn how much pollution is coming from specific sources, measurements are also taken at industrial smokestacks and automobile tailpipes. Pollution is controlled in two ways: with end-of-the-pipe devices that capture pollutants already created and by limiting the quantity of pollutants produced in the first place. End-of-the-pipe devices include catalytic converters in automobiles and various kinds of filters and scrubbers in industrial plants. In a catalytic converter, exhaust gases pass over small beads coated with metals that promote reactions changing harmful substances into less harmful ones. When end-of-the-pipe devices first began to be used, they dramatically reduced pollution at a relatively low cost. As air pollution standards become stricter, it becomes more and more expensive to further clean the air. In order to lower pollution overall, industrial polluters are sometimes allowed to make cooperative deals. For instance, a power company may fulfill its pollution control requirements by investing in pollution control at another plant or factory, where more effective pollution control can be accomplished at a lower cost. End-of-the-pipe controls, however sophisticated, can only do so much. As pollution efforts evolve, keeping the air clean will depend much more on preventing pollution than on curing it. Gasoline, for instance, has been reformulated several times to achieve cleaner burning. Various manufacturing processes have been redesigned so that less waste is produced. Car manufacturers are experimenting with automobiles that run on electricity or on cleaner-burning fuels. Buildings are being designed to take advantage of sun in winter and shade and breezes in summer to reduce the need for artificial heating and cooling, which are usually powered by the burning of fossil fuels. The choices people make in their daily lives can have a significant impact on the state of the air. Using public transportation instead of driving, for instance, reduces pollution by limiting the number of pollution-emitting automobiles on the road. During periods of particularly intense smog, pollution control authorities often urge people to avoid trips by car. To encourage transit use during bad-air periods, authorities in Paris, France, make bus and subway travel temporarily free. Indoor pollution control must be accomplished building by building or even room by room. Proper ventilation mimics natural outdoor air currents, reducing levels of indoor air pollutants by continually circulating fresh air. After improving ventilation, the most effective single step is probably banning smoking in public rooms. Where asbestos has been used in insulation, it can be removed or sealed behind sheathes so that it won’t be shredded and get into the air. Sealing foundations and installing special pipes and pumps can prevent radon from seeping into buildings. On the global scale, pollution control standards are the result of complex negotiations among nations. Typically, developed countries, having already gone through a period of rapid (and dirty) industrialization, are ready to demand cleaner technologies. Less developed nations, hoping for rapid economic growth, are less enthusiastic about pollution controls. They seek lenient deadlines and financial help from developed countries to make the expensive changes necessary to reduce pollutant emissions in their industrial processes. Nonetheless, several important international accords have been reached. In 1988 the United States and 24 other nations agreed in the Long-Range Transboundary Air Pollution Agreement to hold their production of nitrogen oxides, a key contributor to acid rain, to current levels. In the Montreal Protocol on Substances that Deplete the Ozone Layer, adopted in 1987 and strengthened in 1990 and 1992, most nations agreed to stop or reduce the manufacture of CFCs. In 1992 the United Nations Framework Convention on Climate Change negotiated a treaty outlining cooperative efforts to curb global warming. The treaty, which took effect in March 1994, has been legally accepted by 160 of the 165 participating countries. In December 1997 at the Third Conference of the United Nations Framework Convention on Climate Change in Japan, more than 160 nations formally adopted the Kyoto Protocol. This agreement calls for industrialized nations to reduce their emissions of greenhouse gases to levels 5 percent below 1990 emission levels between 2008 and 2012. Negotiators have met regularly since 1995 to iron out the details of how this treaty could be enforced in ways that are agreeable for industrialized countries such as the United States, which releases more greenhouse gases than any other nation, and developing countries that are struggling to become industrialized and often cannot afford the expense that restrictions on greenhouse gas emissions would require. Antipollution measures have helped stem the increase of global pollution emission levels. Between 1970, when the Clean Air Act was passed, and 1995, total emissions of the major air pollutants in the United States decreased by nearly 30 percent. During the same 25-year period, the U. S. population increased 28 percent and vehicle miles traveled increased 116 percent. Air pollution control is a race between the reduction of pollution from each source, such as a factory or a car, and the rapid multiplication of sources. Smog in cities in the United States is expected to increase again as the number of cars and miles driven continues to rise. Meanwhile, developing countries are building up their own industries, and their citizens are buying cars as soon as they can afford them. Ominous changes continue in the global atmosphere. New efforts to control air pollution will be necessary as long as these trends continue.

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.