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Fermat's little theorem

Fermat's little theorem states that if p is a prime number for any integer a, the number ap − a is an integer multiple of p. In the notation of modular arithmetic, this is expressed as a p ≡ a. For example, if a = 2 and p = 7 27 = 128, 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If a is not divisible by p, Fermat's little theorem is equivalent to the statement that ap − 1 − 1 is an integer multiple of p, or in symbols: a p − 1 ≡ 1. For example, if a = 2 and p = 7 26 = 64, 64 − 1 = 63 = 7 × 9 is thus a multiple of 7. Fermat's little theorem is the basis for the Fermat primality test and is one of the fundamental results of elementary number theory; the theorem is named after Pierre de Fermat, who stated it in 1640. It is called the "little theorem" to distinguish it from Fermat's last theorem. Pierre de Fermat first stated the theorem in a letter dated October 18, 1640, to his friend and confidant Frénicle de Bessy, his formulation is equivalent to the following: If p is a prime and a is any integer not divisible by p a p − 1 − 1 is divisible by p. Fermat's original statement was Tout nombre premier mesure infailliblement une des puissances – 1 de quelque progression que ce soit, et l'exposant de la dite puissance est sous-multiple du nombre premier donné – 1.

This may be translated, with explanations and formulas added in brackets for easier understanding, as: Every prime number divides one of the powers minus one of any progression, the exponent of this power divides the given prime minus one. After one has found the first power that satisfies the question, all those whose exponents are multiples of the exponent of the first one satisfy the question. Fermat did not consider the case where a is a multiple of p nor prove his assertion, only stating: Et cette proposition est généralement vraie en toutes progressions et en tous nombres premiers. Euler provided the first published proof in 1736, in a paper titled "Theorematum Quorundam ad Numeros Primos Spectantium Demonstratio" in the Proceedings of the St. Petersburg Academy, but Leibniz had given the same proof in an unpublished manuscript from sometime before 1683; the term "Fermat's little theorem" was first used in print in 1913 in Zahlentheorie by Kurt Hensel: Für jede endliche Gruppe besteht nun ein Fundamentalsatz, welcher der kleine Fermatsche Satz genannt zu werden pflegt, weil ein ganz spezieller Teil desselben zuerst von Fermat bewiesen worden ist.

An early use in English occurs in A. A. Albert's Modern Higher Algebra, which refers to "the so-called'little' Fermat theorem" on page 206; some mathematicians independently made the related hypothesis only if p is prime. Indeed, the "if" part is true, it is a special case of Fermat's little theorem. However, the "only if" part is false: For example, 2341 ≡ 341 = 11 × 31 is a pseudoprime. See below. Several proofs of Fermat's little theorem are known, it is proved as a corollary of Euler's theorem. Euler's theorem is a generalization of Fermat's little theorem: for any modulus n and any integer a coprime to n, one has a φ ≡ 1, where φ denotes Euler's totient function. Fermat's little theorem is indeed a special case, because if n is a prime number φ = n − 1. A corollary of Euler's theorem is: for every positive integer n, if the integer a is coprime with n x ≡ y implies a x ≡ a y, for any integers x and y; this follows from Euler's theorem, since, if x ≡ y x = y + k φ for some integer k, one has a x = a y

Battle of Big Bethel

The Battle of Big Bethel was one of the earliest land battles of the American Civil War. It took place on the Virginia Peninsula, near Newport News, on June 10, 1861. Virginia's decision to secede from the Union had been ratified by popular vote on May 23, Confederate Col. John B. Magruder was sent down the peninsula to deter any advance on the state capital Richmond by Union troops based at the well-defended post of Fort Monroe; this garrison was commanded by Maj Gen. Benjamin Butler, a former Massachusetts lawyer and politician, who established a new camp at nearby Hampton and another at Newport News. Magruder had established two camps, within range of the Union lines, at Big Bethel and Little Bethel, as a lure to draw his opponent into a premature action. Butler took the bait, when he and an aide, Maj. Theodore Winthrop, devised a plan for a dawn attack on June 10, after a night march to drive the Confederates back from their bases. Butler chose not to lead the force in person, for which he was criticized.

The plan proved too complex for his poorly-trained subordinates to carry out at night, his staff had omitted to communicate the passwords. They were trying to advance without knowledge of the layout or strength of the Confederate positions, when a friendly fire incident gave away their own; the commander in the field, Massachusetts militia Gen. Ebenezer Peirce, received most of the blame for the failed operation; the Union forces suffered 76 casualties, with 18 killed, including Maj. Winthrop and Lt. John T. Greble, the first regular army officer killed in the war; the Confederates suffered only eight casualties, with one killed. Although Magruder subsequently withdrew to Yorktown and his defensive line along the Warwick River, he had won a propaganda victory and local Union forces attempted no further significant advance until the Peninsula Campaign of 1862. While small in comparison to many battles, Big Bethel attracted exaggerated importance because of the general feeling that the war would be short.

The engagement was known as the Battle of Bethel Church or Great Bethel. After the American Civil War began with the formal surrender of Fort Sumter to Confederate forces on April 14, 1861, President Lincoln's call for volunteers to suppress the rebellion on April 15, 1861, Virginia's political leaders set in motion the process of seceding from the Union and joining the Confederacy. Before secession was formally accomplished, Virginia agreed to coordinate its state military forces with the Confederacy and began to seize federal property; the United States Regular Army garrison under the command of Col. Justin Dimick held Fort Monroe, a nearly impregnable fortress at Old Point Comfort on the southern tip of the Virginia Peninsula between the York River and the James River where they empty into Chesapeake Bay; the bay was to the east of the fort and Hampton Roads was to the south. The fort was supported by the Union Navy at Hampton Roads and could be reinforced and resupplied by water without attack by shore batteries or harassment by the nearly non-existent Virginia or Confederate naval forces.

The fort was nearly immune from attack from the land side since it could be approached only over a narrow causeway and a narrow isthmus and had massive walls and hundreds of cannons. An inlet called Mill Creek was the body of water that cut the fort off from mainland of the Peninsula. Col. Dimick refused to surrender the fort and the small and poorly equipped Virginia militia forces in the area had no hope of taking the fort by force after April 20, 1861, when the small Union garrison was reinforced by two Massachusetts volunteer regiments within a few days of the Virginia convention voting to secede from the Union on April 17, 1861; this important fort would provide a base for the blockade of Norfolk and the Chesapeake Bay and for the recovery of southeast Virginia and the Virginia Peninsula for the Union. Because Massachusetts militia forces were ready to respond to President Lincoln's call for volunteers, two 90–day regiments, the 3rd Massachusetts Militia commanded by Col. David W. Wardrop and the 4th Massachusetts Militia commanded by Col. Abner B.

Packard, were able to reinforce Fort Monroe's garrison of 415 regulars within five days of the President's call. The 4th Massachusetts was the first to arrive; these reinforcements helped assure that this strong point and base of operations would be prepared for defense and saved for the Union. On May 13, 1861, the 1st Regiment, Vermont Volunteer Infantry under Col. J. Wolcott Phelps joined the garrison and several other volunteer regiments from New York soon followed. On May 14, 1861 while Col. Dimick was still in command of the garrison, he seized a well just outside the fort in what was Elizabeth City County because the fort did not have enough water for its original small garrison, his forces occupied the Mill Creek Bridge, needed for access to the Peninsula from the fort and the nearby Clark farm. The fort soon could not hold all of the arriving reinforcements so Union forces established Camp Troy, soon renamed Camp Hamilton in honor of an aide to General-in-Chief Winfield Scott, on the Segar farm on the Hampton side of Mill Creek, within range of the guns of Fort Monroe.

While Col. Dimick remained in command of the 415 regular army soldiers, Volunteer Maj. Gen. Benjamin F. Butler of Massachusetts took command of Fort Monroe and the entire garrison on May 23, 1861. With continuing reinforcements, Butler could not only retain the Union hold on Fort Monroe but support the Union blockade of Chesapeake Bay, move up the Peninsula and threaten to retake Norfolk and other locations on the south side of Hampton Roads from the Confederates. On May 27, 1861, Gen. Butler sent

Thekna mine

The Thekna mine is a large mine located in central Albania in Dibër County, 40 kilometres east of the capital, Tirana. Thekna represents one of the largest chromium reserve in Albania and one of the largest in Europe having estimated reserves of 0.652 million tonnes of ore grading 38% chromium metal. The mine is part of the Bulqizë Massif, a 370 square kilometres area which has a rock thickness between 4 kilometres and 6 kilometres and contains 65 verified chromium deposits and occurrences; the deposit has been explored to depths of up to 1,560 metres and the geological reserves amount to 12 million tonnes of which 7.5 million tonnes grading over 38% chromium metal. The Thekna mine began operating in 1959. Between 1957 and 1959 it produced chromium ore only from occurrences. In 1959 production from the mine started; the total combined chromium ore production from the mine between 1959 and 2006 amounted to 1,198,000 tonnes. The deepest level of the mine is the Descenderia No. 7. The chromium ore reserves of the mine are split into two categories above and below Descenderia No.7.

The proven ore reserves located above the Descenderia No.7 amount to 652,300 tonnes of ore grading 38% chromium metal. The proven ore reserves located below the Descenderia No.7 are not estimated or calculated but are expected to be grading 45% chromium metal. The mine's total reserves amount to 0.652 million tonnes of ore grading 38% chromium metal. The mine was privatised in 1999

Pacifism in Germany

The existence of pacifism in Germany has changed over time, with the consistent feature of having diverse groups with a shared belief in an opposition to participating in war. These movements both individually and collectively, have been small in their numbers and have not been well organised. With a culture of war in the early history of Germany, pacifism was not a culturally significant group; this was driven by the government as they attempted to use the media in order to promote the expansion of Germany as a growing empire. The exception to this is during the Cold War with the Bonn demonstration with a large turnout of around 300,000 people. Christian peace groups have been the most consistent groups within the classification of pacifists as an opposition to violence is a key part of their faith; the size, whilst remaining small varies over the history of the Federal Republic of Germany. The reception from the public regarding pacifists changes depending on the historical period. With the unification of Germany as a single state, the country began to expand militarily as an international power, which in turn created a pacifist movement in Germany.

This first movement was called the German Peace Society and was founded in 1892. However, the movement was small with only 10,000 active members at its peak, as it did not resonate with the wider population, in favour of the German expansion. With low support the ability of the movement to grow was limited to the ability to speak, limited due to the government declaring a state of war; the rapid military expansion of Germany was largely popular with the population. The movement, prior to the outbreak of war began to decline in support for the movement as the public support of a war grew and avoid a response from the government. During this time there were two Hague Peace Conferences, these conferences resulted in various multi-lateral treaties regarding military expansion and foreign policy; the conference was about establishing the universal values and therefore the obligations of states. In 1899 there was a Hauge Peace Conference, Germany did not attend, it was not until the 1907 conference where Germany would participate.

At this conference Germany was considered uncooperative, due to the restrictions on their ability to expand their military power. As Germany was a rising power there was a hesitation to reduce their expansion as it would restrict their self-defence capabilities. German Pacifism was not as organised in this era. During this period a group of female war opponents emerged, a pacifist group who were opposed to the war as it was, according to this group, caused by masculine values and attitudes; this group overlapped with the group of advocates for women’s’ rights during the same period, as well as the socialist movement. This activism from female groups was the result of changing cultural and social roles of women which had developed during the war. Early in the war attitudes towards war were positive, this along with the conscription of German soldiers resulted in the pacifists’ movement remaining a small group; as the country was in a state of war, the government was involved in censorship of the population, so the pacifist publications were censored to not have views which directly opposed the government, rather they would be written in order to talk about the wider issue of war with other countries having the primary role at the beginning of war.

The government remained lenient towards those with pacifist attitudes which contrasts the government attitude of the Nazi Government and other governments which enacted the concept of total war on the home front. In 1915 at the Hague, the Women's Peace Conference met to discuss methods to end the war through negotiations. Attending this event was four German delegates. After this conference, the government began to restrict pacifists and their groups as there was a consensus that this conference had weakened the position of Germany for negotiations. Towards the end of the war, the public support massively declines, which resulted in an increase in the support for pacifist ideas, despite this the movement remained small in size. At this time the government tried to assert more control over the home front in order to provide more resources for soldiers on the front line. With the Russian Revolution, the pacifist movement gained support and inspiration through the Marxist ideology behind the revolution.

The literature during the war was restricted and the literature, promoted all displayed similar views. These views support the war effort, achieved through the justification of the war, as well as the celebration of acts of heroism and sacrifice. Pacifist literature was non-existent during this time period which continued until the end of the war before it became more discussed in the literature; the existence of pacifists in Germany is at its lowest and least organised during this time, as a result of the Nazi Government’s policies regarding movements which oppose their regime. Pacifists during this time are individuals, who may not be a part of a formalised group, rather they act out the ideology of pacifism. A key reason for this was the continued support of the war effort as well as public support for the Fuhrer of Germany, Adolf Hitler remaining high throughout the war; this period was when popular support for pacifism was at its lowest as there was significant support for the war effort.

As Nazism was built on the outcome of the First World War, there was significant pro-war sentiment in the ideology as a means of resolving the issues

Oakland, Maryland

For other places with the same name, see Oakland. Oakland is a town in the west-central part of Garrett County, United States; the town has a population of 1,925 according to the 2010 United States Census. The town is the county seat of Garrett County and is located within the Pittsburgh DMA. Oakland is situated only miles from the source of the Potomac River, which flows directly into Chesapeake Bay, it is near the Wisp Resort at Deep Creek Lake, a major ski resort for many Marylanders and other visitors. Oakland was formally incorporated as a town in 1862; the town is home to a historic B&O railroad station, listed on the National Register of Historic Places in 1973, restored in the 2000s. Trains still run on the rail tracks behind the station, but it is used for special organizations or gatherings at present. A gift shop is located within the station. In front of the station, there are a plethora of festivities that go on seasonal activities such as housing the town Christmas tree, decorating the plaza for a holiday, sometimes parties.

Main Street of Oakland consists of historic two to four story edifices that house the main shopping facilities in the area, such as a theatre, book store, a local pharmacy, antique shops, clothing stores and banks. Many of the homes and businesses in the downtown area are examples of Victorian architecture. Much of the central section of Oakland is part of the Oakland Historic District, listed on the National Register of Historic Places in 1984. Listed on the National Register are the Garrett County Courthouse and Hoye Site. One of the most prominent and historic churches in Oakland is St. Matthew's Episcopal Church, where U. S. Presidents Ulysses S. Grant, James Garfield, Grover Cleveland, Benjamin Harrison have all attended services; because of this, it is now called the "Church of Presidents." Another prominent and historic church is St. Peter the Apostle Church, a Catholic church located on Fourth Street. A large neoclassical courthouse is very prominent and dominates the town center. In the late 19th century and early 20th century, a large hotel named the Oakland Hotel was located near the downtown railroad station.

It was constructed in 1878 by the B&O Railroad. The hotel was a major tourist attraction for that time period until it was torn down in the early 20th century. Oakland is in the south-central to western portion of Garrett County, located at 39°24′38″N 79°24′16″W, it is set in a small valley. According to the United States Census Bureau, the town has a total area of 2.60 square miles, of which 2.59 square miles is land and 0.01 square miles is water. It is only 6.61 miles from Second Street to Deep Creek Lake. Oakland, owing to its high elevation and valley location, is among the coldest and snowiest locales in the state of Maryland, has a warm-summer humid continental climate, with 106 inches or 2.69 metres of snowfall in an average season. The monthly mean temperature ranges from 25.1 °F in January to 68.4 °F in July, with temperatures not reaching above freezing on an average 34 afternoons and falling to 0 °F or below on an average of 5.8 mornings. The average first and last dates for freezing temperatures are September 28 and May 15, respectively.

The state record low of −40 °F or −40 °C was recorded here on January 13, 1912. The most snow in 24 hours was 40.0 inches on February 16, 1908. According to weather data tallied between July 1, 1985 and June 30, 2015 for every location in the National Oceanic and Atmospheric Administration's official climate database, Oakland is the snowiest place in the state of Maryland with an average of 105.9 inches or 2.69 metres of snow per year. As of the census of 2010, there were 1,925 persons, 875 households, 470 families living in the town; the population density was 743.2 inhabitants per square mile. There were 1,009 housing units at an average density of 389.6 per square mile. The racial makeup of the town was 98.0% White, 0.2% African American, 0.3% Native American, 0.6% Asian, 0.1% from other races, 0.9% from two or more races. Hispanic or Latino of any race were 0.7% of the population. There were 875 households of which 23.1% had children under the age of 18 living with them, 40.1% were married couples living together, 10.5% had a female householder with no husband present, 3.1% had a male householder with no wife present, 46.3% were non-families.

40.5% of all households were made up of individuals and 16.6% had someone living alone, 65 years of age or older. The average household size was 2.03 and the average family size was 2.73. The median age in the town was 46.9 years. 17.5% of residents were under the age of 18. The gender makeup of the town was 52.3 % female. As of the census of 2000, there were 1,930 persons, 787 households, 447 families living in the town; the population density was 915.7 inhabitants per square mile. There were 918 housing units at an average density of 435.5 per square mile. The racial makeup of the town was 98.13% White, 0.73% African American, 0.16% Native American, 0.57% Asian, 0.16% from other races, 0.26% from two or more races. Hispanic or Latino of any race were 0.78% of the population. 33% of Oakland's residents were of


The Muckleshoot are a Lushootseed-speaking Indian tribe, part of the Coast Salish peoples of the Pacific Northwest. They are descendants of the Duwamish and Puyallup peoples whose traditional territory was located along the Green and White rivers, including up to the headwaters in the foothills of the Cascade Mountains, in present-day Washington State. Since the mid-19th century, their reservation is located in the area of Auburn, about 15 miles northeast of the port of Tacoma and 35 miles southeast of Seattle, another major port; the federally recognized Muckleshoot Indian Tribe is a group that formed post-Treaty, made up of related peoples who shared territory and a reservation near Auburn. They organized a government in 1936; these include the following: Buklshuhls - they lived along the White River, from present-day Kent eastwards to the mountains and to the Green River Duwamish - this people formed two bands before the mid-1850s Dxʷ'Dəw? Abš / Dkhw'Duw'Absh Xacuabš Snoqualmie - they lived along the Tolt River and the Snoqualmie River) Upper Puyallup people: Puyallup bands along the Upper Puyallup River White River Valley tribes:Stkamish / Skekomish Smulkamish / Smalhkamish - They lived in villages on the present Muckleshoot Indian Reservation and near present-day Enumclaw) Skopamish - They lived in the central Green River Valley above the former confluence near present-day Auburn.

The term skop means "first big and little," in apparent reference to fluctuations of the Green River. Another source says their name is derived from the village name ill-AHL-koh at the historic confluence of the White and Green rivers at the present-day town of Auburn from the striped appearance of the Green River below the confluence before the waters merged. Tkwakwamish / T'Qua-qua-mish Yilalkoamish tribe Dothliuk Traditionally, the ancestors of the Muckleshoot lived along the eastern shores of Washington State's Puget Sound region and the adjacent rivers of the Cascade Range, they spoke a local form of Lushootseed. Most Muckleshoot today do not speak their ancestral language; the tribe has an active program for its resuscitation. Most Muckleshoot now live near the 15.871 km ² Muckleshoot Reservation. They have an approximate population of more than 3,000, making the Muckleshoot one of the largest Native American tribes in Washington State; the 2000 census reported a resident population of 3,606 on reservation land, with 28.65 percent reported Native American heritage.

The Coast Salish and Muckleshoot had long absorbed other peoples into their tribes and have had multi-racial descendants. Their children are raised culturally as Muckleshoot; the reservation is located on Muckleshoot Prairie, between the White and Green rivers southeast of the city of Auburn in King and Pierce counties. The city of Auburn extends within the reservation; some 72.6 percent of the reservation's population lives within the city boundaries. Although they were skilled hunters, salmon fishing was the mainstay of traditional Coast Salish life; the people gathered and cured salmon, sometimes trading it with other peoples along the coast and inland. Because it was central to survival, salmon was treated with reverence. In the elaborate First Salmon Ceremony, still observed, the entire community shares the flesh of a Spring Chinook, they return its remains to the river. This is; the other ceremony for the first salmon is to roast it. The Muckleshoot toss the bones and ashes back into the water or stream where they took the salmon, believing that the fish would come alive again.

With a endless supply of food, the people could engage in various crafts, including weaving, wood-carving, basket-making. A complex social structure emerged, consisting of a nobility, middle class, slaves; the latter were captured members of other tribes taken in raids or warfare. Coast Salish life changed radically as a result of first enc