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140-518: Integers can be considered either in themselves or as solutions to equations ( Diophantine geometry ). Questions in number theory are often best understood through the study of analytical objects (for example, the Riemann zeta function ) that encode properties of the integers, primes or other number-theoretic objects in some fashion ( analytic number theory ). One may also study real numbers in relation to rational numbers; for example, as approximated by

280-658: A 1 mod m 1 {\displaystyle n\equiv a_{1}{\bmod {m}}_{1}} , n ≡ a 2 mod m 2 {\displaystyle n\equiv a_{2}{\bmod {m}}_{2}} could be solved by a method he called kuṭṭaka , or pulveriser ; this is a procedure close to (a generalisation of) the Euclidean algorithm, which was probably discovered independently in India. Āryabhaṭa seems to have had in mind applications to astronomical calculations. Brahmagupta (628 AD) started

420-488: A Diophantine equations a polynomial equations to which rational or integer solutions are sought. While Greek astronomy probably influenced Indian learning, to the point of introducing trigonometry , it seems to be the case that Indian mathematics is otherwise an indigenous tradition; in particular, there is no evidence that Euclid's Elements reached India before the 18th century. Āryabhaṭa (476–550 AD) showed that pairs of simultaneous congruences n ≡

560-445: A hypersurface , and simultaneous Diophantine equations give rise to a general algebraic variety V over K ; the typical question is about the nature of the set V ( K ) of points on V with co-ordinates in K , and by means of height functions , quantitative questions about the "size" of these solutions may be posed, as well as the qualitative issues of whether any points exist, and if so whether there are an infinite number. Given

700-435: A metropolitan area with a population of over 20 million. On 17 November 1997, 62 people, mostly tourists, were massacred near Luxor . In late February 2005, Mubarak announced a reform of the presidential election law, paving the way for multi-candidate polls for the first time since the 1952 movement . However, the new law placed restrictions on the candidates, and led to Mubarak's easy re-election victory. Voter turnout

840-597: A republic . For a brief period between 1958 and 1961 Egypt merged with Syria to form the United Arab Republic . Egypt fought several armed conflicts with Israel in 1948 , 1956 , 1967 and 1973 , and occupied the Gaza Strip intermittently until 1967. In 1978, Egypt signed the Camp David Accords , which recognised Israel in exchange for its withdrawal from the occupied Sinai. After

980-512: A "table", for example, with a view to applications. It is not known what these applications may have been, or whether there could have been any; Babylonian astronomy , for example, truly came into its own only later. It has been suggested instead that the table was a source of numerical examples for school problems. While evidence of Babylonian number theory is only survived by the Plimpton 322 tablet, some authors assert that Babylonian algebra

1120-611: A cash-crop monoculture before the end of the century, concentrating land ownership and shifting production towards international markets. Muhammad Ali annexed Northern Sudan (1820–1824), Syria (1833), and parts of Arabia and Anatolia ; but in 1841 the European powers, fearful lest he topple the Ottoman Empire itself, forced him to return most of his conquests to the Ottomans. His military ambition required him to modernise

1260-618: A deputy to Egypt to govern for him. In 828 another Egyptian revolt broke out, and in 831 the Copts joined with native Muslims against the government. Eventually the power loss of the Abbasids in Baghdad led for general upon general to take over rule of Egypt, yet being under Abbasid allegiance, the Tulunid dynasty (868–905) and Ikhshidid dynasty (935–969) were among the most successful to defy

1400-455: A first foray towards both Évariste Galois 's work and algebraic number theory . Starting early in the nineteenth century, the following developments gradually took place: Algebraic number theory may be said to start with the study of reciprocity and cyclotomy , but truly came into its own with the development of abstract algebra and early ideal theory and valuation theory; see below. A conventional starting point for analytic number theory

1540-776: A founding member of the United Nations, the Non-Aligned Movement , the Arab League , the African Union , Organisation of Islamic Cooperation , World Youth Forum , and a member of BRICS . The English name "Egypt" is derived from the Ancient Greek " Aígyptos " (" Αἴγυπτος "), via Middle French "Egypte" and Latin " Aegyptus ". It is reflected in early Greek Linear B tablets as "a-ku-pi-ti-yo". The adjective "aigýpti-"/"aigýptios"

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1680-405: A list of " Pythagorean triples ", that is, integers ( a , b , c ) {\displaystyle (a,b,c)} such that a 2 + b 2 = c 2 {\displaystyle a^{2}+b^{2}=c^{2}} . The triples are too many and too large to have been obtained by brute force . The heading over the first column reads: "The takiltum of

1820-507: A military unit to be reckoned with. The resentment for the military way of life eventually faded from the men and a new ideology took hold, one of nationalism and pride. It was with the help of this newly reborn martial unit that Muhammad Ali imposed his rule over Egypt. The policy that Mohammad Ali Pasha followed during his reign explains partly why the numeracy in Egypt compared to other North-African and Middle-Eastern countries increased only at

1960-406: A naïve sense, analogous to primes among the integers.) The initial impetus for the development of ideal numbers (by Kummer ) seems to have come from the study of higher reciprocity laws, that is, generalisations of quadratic reciprocity . Number fields are often studied as extensions of smaller number fields: a field L is said to be an extension of a field K if L contains K . (For example,

2100-471: A non-elementary one. Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, the proofs of these results are not particularly accessible, in part because the range of tools they use is, if anything, unusually broad within mathematics. Analytic number theory may be defined Some subjects generally considered to be part of analytic number theory, for example, sieve theory , are better covered by

2240-636: A projective variety that has extra points at infinity . The general approach of Diophantine geometry is illustrated by Faltings's theorem (a conjecture of L. J. Mordell ) stating that an algebraic curve C of genus g > 1 over the rational numbers has only finitely many rational points . The first result of this kind may have been the theorem of Hilbert and Hurwitz dealing with the case g = 0. The theory consists both of theorems and many conjectures and open questions. Egypt Egypt ( Arabic : مصر Miṣr [mesˁr] , Egyptian Arabic pronunciation: [mɑsˤr] ), officially

2380-673: A remark on homogeneous equations f = 0 over the rational field , attributed to C. F. Gauss , that non-zero solutions in integers (even primitive lattice points) exist if non-zero rational solutions do, and notes a caveat of L. E. Dickson , which is about parametric solutions. The Hilbert – Hurwitz result from 1890 reducing the Diophantine geometry of curves of genus 0 to degrees 1 and 2 ( conic sections ) occurs in Chapter 17, as does Mordell's conjecture . Siegel's theorem on integral points occurs in Chapter 28. Mordell's theorem on

2520-618: A remarkably small rate, as investment in further education only took place in the military and industrial sector. Muhammad Ali was succeeded briefly by his son Ibrahim (in September 1848), then by a grandson Abbas I (in November 1848), then by Said (in 1854), and Isma'il (in 1863) who encouraged science and agriculture and banned slavery in Egypt. Egypt under the Muhammad Ali dynasty remained nominally an Ottoman province. It

2660-522: A report alleging that Egypt had become an international centre for torture, where other nations send suspects for interrogation, often as part of the War on Terror . Egypt's foreign ministry quickly issued a rebuttal to this report. Constitutional changes voted on 19 March 2007 prohibited parties from using religion as a basis for political activity, allowed the drafting of a new anti-terrorism law, authorised broad police powers of arrest and surveillance, and gave

2800-721: A sovereign union known as the United Arab Republic . The union was short-lived, ending in 1961 when Syria seceded. During most of its existence, the United Arab Republic was also in a loose confederation with North Yemen (or the Mutawakkilite Kingdom of Yemen), known as the United Arab States . In the early 1960s, Egypt became fully involved in the North Yemen Civil War . Despite several military moves and peace conferences,

2940-405: A tendency to clothe the old results, their extensions, and proofs in the new geometrical language. Sometimes, however, the full implications of results are best described in a geometrical setting. Lang has these aspects very much in mind in this book, and seems to miss no opportunity for geometric presentation. This accounts for his title "Diophantine Geometry." He notes that the content of the book

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3080-403: Is Dirichlet's theorem on arithmetic progressions (1837), whose proof introduced L-functions and involved some asymptotic analysis and a limiting process on a real variable. The first use of analytic ideas in number theory actually goes back to Euler (1730s), who used formal power series and non-rigorous (or implicit) limiting arguments. The use of complex analysis in number theory comes later:

3220-485: Is a fixed rational number whose square root is not rational.) For that matter, the 11th-century chakravala method amounts—in modern terms—to an algorithm for finding the units of a real quadratic number field. However, neither Bhāskara nor Gauss knew of number fields as such. The grounds of the subject were set in the late nineteenth century, when ideal numbers , the theory of ideals and valuation theory were introduced; these are three complementary ways of dealing with

3360-440: Is an algebraic number. Fields of algebraic numbers are also called algebraic number fields , or shortly number fields . Algebraic number theory studies algebraic number fields. Thus, analytic and algebraic number theory can and do overlap: the former is defined by its methods, the latter by its objects of study. It could be argued that the simplest kind of number fields (viz., quadratic fields) were already studied by Gauss, as

3500-500: Is credited to the early Pythagoreans (pre- Theodorus ). By revealing (in modern terms) that numbers could be irrational, this discovery seems to have provoked the first foundational crisis in mathematical history; its proof or its divulgation are sometimes credited to Hippasus , who was expelled or split from the Pythagorean sect. This forced a distinction between numbers (integers and the rationals—the subjects of arithmetic), on

3640-564: Is directly cognate with the Biblical Hebrew Miṣráyīm (מִצְרַיִם / מִצְרָיִם), meaning "the two straits", a reference to the predynastic separation of Upper and Lower Egypt . Also mentioned in several Semitic languages as Mesru , Misir and Masar . The oldest attestation of this name for Egypt is the Akkadian "mi-iṣ-ru" ("miṣru") related to miṣru/miṣirru/miṣaru , meaning "border" or "frontier". The Neo-Assyrian Empire used

3780-404: Is even", "if an odd number measures [= divides] an even number, then it also measures [= divides] half of it"), but it is all that is needed to prove that 2 {\displaystyle {\sqrt {2}}} is irrational . Pythagorean mystics gave great importance to the odd and the even. The discovery that 2 {\displaystyle {\sqrt {2}}} is irrational

3920-478: Is largely versions of the Mordell–Weil theorem , Thue–Siegel–Roth theorem , Siegel's theorem, with a treatment of Hilbert's irreducibility theorem and applications (in the style of Siegel). Leaving aside issues of generality, and a completely different style, the major mathematical difference between the two books is that Lang used abelian varieties and offered a proof of Siegel's theorem, while Mordell noted that

4060-517: Is part of the broader field of arithmetic geometry . Four theorems in Diophantine geometry that are of fundamental importance include: Serge Lang published a book Diophantine Geometry in the area in 1962, and by this book he coined the term "Diophantine geometry". The traditional arrangement of material on Diophantine equations was by degree and number of variables , as in Mordell 's Diophantine Equations (1969). Mordell's book starts with

4200-627: Is the Classical Quranic Arabic and modern official name of Egypt, while " Maṣr " ( Egyptian Arabic pronunciation: [mɑsˤɾ] ; مَصر ) is the local pronunciation in Egyptian Arabic . The current name of Egypt, Misr/Misir/Misru, stems from the Ancient Semitic name for it. The term originally connoted " Civilization " or " Metropolis ". Classical Arabic Miṣr (Egyptian Arabic Maṣr )

4340-585: Is the capital and largest city of Egypt , while Alexandria , the second-largest city, is an important industrial and tourist hub at the Mediterranean coast . At approximately 110 million inhabitants, Egypt is the 14th-most populated country in the world , and the third-most populated in Africa. Egypt has one of the longest histories of any country, tracing its heritage along the Nile Delta back to

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4480-403: Is through one of Plato's dialogues—namely, Theaetetus —that it is known that Theodorus had proven that 3 , 5 , … , 17 {\displaystyle {\sqrt {3}},{\sqrt {5}},\dots ,{\sqrt {17}}} are irrational. Theaetetus was, like Plato, a disciple of Theodorus's; he worked on distinguishing different kinds of incommensurables , and

4620-408: Is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions , which are generalizations of the Riemann zeta function , a key analytic object at

4760-499: Is to determine when a Diophantine equation has solutions, and if it does, how many. The approach taken is to think of the solutions of an equation as a geometric object. Diophantine geometry In mathematics , Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry . By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations. Diophantine geometry

4900-690: The 1948 Arab–Israeli War . During the 1967 war, an Emergency Law was enacted, and remained in effect until 2012, with the exception of an 18-month break in 1980/81. Under this law, police powers were extended, constitutional rights suspended and censorship legalised. At the time of the fall of the Egyptian monarchy in the early 1950s, less than half a million Egyptians were considered upper class and rich, four million middle class and 17 million lower class and poor. Fewer than half of all primary-school-age children attended school, most of them being boys. Nasser's policies changed this. Land reform and distribution,

5040-611: The Anglo-Egyptian Treaty was concluded and British troops withdrew from Egypt, except for the Suez Canal. The treaty did not resolve the question of Sudan , which, under the terms of the existing Anglo-Egyptian Condominium Agreement of 1899, stated that Sudan should be jointly governed by Egypt and Britain, but with real power remaining in British hands. Britain used Egypt as a base for Allied operations throughout

5180-687: The Arab League , but it was supported by most Egyptians. Sadat was assassinated by an Islamic extremist in October 1981. Hosni Mubarak came to power after the assassination of Sadat in a referendum in which he was the only candidate. He became another leader to dominate the Egyptian history . Hosni Mubarak reaffirmed Egypt's relationship with Israel yet eased the tensions with Egypt's Arab neighbours. Domestically, Mubarak faced serious problems. Mass poverty and unemployment led rural families to stream into cities like Cairo where they ended up in crowded slums, barely managing to survive. On 25 February 1986 ,

5320-700: The Arab Republic of Egypt , is a transcontinental country spanning the northeast corner of Africa and the Sinai Peninsula in the southwest corner of Asia . It is bordered by the Mediterranean Sea to the north , the Gaza Strip of Palestine and Israel to the northeast , the Red Sea to the east, Sudan to the south , and Libya to the west . The Gulf of Aqaba in the northeast separates Egypt from Jordan and Saudi Arabia . Cairo

5460-516: The Arab Spring , which led to the 2011 Egyptian revolution and overthrow of Hosni Mubarak , the country faced a protracted period of political unrest ; this included the election in 2012 of a brief, short-lived Muslim Brotherhood -aligned Islamist government spearheaded by Mohamed Morsi , and its subsequent overthrow after mass protests in 2013 . The current government, a semi-presidential republic led by Abdel Fattah el-Sisi since he

5600-517: The Arithmetica , was translated into Arabic by Qusta ibn Luqa (820–912). Part of the treatise al-Fakhri (by al-Karajī , 953 – ca. 1029) builds on it to some extent. According to Rashed Roshdi, Al-Karajī's contemporary Ibn al-Haytham knew what would later be called Wilson's theorem . Other than a treatise on squares in arithmetic progression by Fibonacci —who traveled and studied in north Africa and Constantinople—no number theory to speak of

5740-706: The Ottoman Turks , Egyptian Mamluks who had ruled Egypt for centuries, and Albanian mercenaries in the service of the Ottomans. After the French were expelled, power was seized in 1805 by Muhammad Ali Pasha , an Albanian military commander of the Ottoman army in Egypt. Muhammad Ali massacred the Mamluks and established a dynasty that was to rule Egypt until the revolution of 1952. The introduction in 1820 of long-staple cotton transformed its agriculture into

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5880-457: The Ptolemaic line was Cleopatra VII , who committed suicide following the burial of her lover Mark Antony , after Octavian had captured Alexandria and her mercenary forces had fled. The Ptolemies faced rebellions of native Egyptians and were involved in foreign and civil wars that led to the decline of the kingdom and its annexation by Rome. Christianity was brought to Egypt by Saint Mark

6020-631: The Ptolemaic dynasty . The Ptolemaic Kingdom was a powerful Hellenistic state, extending from southern Syria in the east, to Cyrene to the west, and south to the frontier with Nubia. Alexandria became the capital city and a centre of Greek culture and trade. To gain recognition by the native Egyptian populace, they named themselves as the successors to the Pharaohs. The later Ptolemies took on Egyptian traditions, had themselves portrayed on public monuments in Egyptian style and dress, and participated in Egyptian religious life. The last ruler from

6160-632: The Third Dynasty pyramid of Djoser and the Fourth Dynasty Giza pyramids . The First Intermediate Period ushered in a time of political upheaval for about 150 years. Stronger Nile floods and stabilisation of government, however, brought back renewed prosperity for the country in the Middle Kingdom c.  2040 BCE, reaching a peak during the reign of Pharaoh Amenemhat III . A second period of disunity heralded

6300-590: The Wafd Party led the Egyptian nationalist movement to a majority at the local Legislative Assembly. When the British exiled Zaghlul and his associates to Malta on 8 March 1919, the country arose in its first modern revolution . The revolt led the UK government to issue a unilateral declaration of Egypt's independence on 22 February 1922. Following independence from the United Kingdom, Sultan Fuad I assumed

6440-709: The early modern period (17th to early 19th centuries). The Chinese remainder theorem appears as an exercise in Sunzi Suanjing (3rd, 4th or 5th century CE). (There is one important step glossed over in Sunzi's solution: it is the problem that was later solved by Āryabhaṭa 's Kuṭṭaka – see below .) The result was later generalized with a complete solution called Da-yan-shu ( 大衍術 ) in Qin Jiushao 's 1247 Mathematical Treatise in Nine Sections which

6580-658: The elections of June 2014 El-Sisi won with a percentage of 96.1%. On 8 June 2014, Abdel Fatah el-Sisi was officially sworn in as Egypt's new president. Under President el-Sisi, Egypt has implemented a rigorous policy of controlling the border to the Gaza Strip, including the dismantling of tunnels between the Gaza strip and Sinai. In April 2018, El-Sisi was re-elected by a landslide in election with no real opposition. In April 2019, Egypt's parliament extended presidential terms from four to six years. President Abdel Fattah al-Sisi

6720-408: The 1990s an Islamist group, Al-Gama'a al-Islamiyya , engaged in an extended campaign of violence, from the murders and attempted murders of prominent writers and intellectuals, to the repeated targeting of tourists and foreigners. Serious damage was done to the largest sector of Egypt's economy—tourism —and in turn to the government, but it also devastated the livelihoods of many of the people on whom

6860-532: The 6th–4th millennia BCE. Considered a cradle of civilisation , the Ancient Egypt saw some of the earliest developments of writing, agriculture, urbanisation, organised religion and central government. Egypt was an early and important centre of Christianity , later adopting Islam from the seventh century onwards. Cairo became the capital of the Fatimid Caliphate in the tenth century, and of

7000-629: The Abbasid Caliph. Muslim rulers remained in control of Egypt for the next six centuries, with Cairo as the seat of the Fatimid Caliphate . With the end of the Ayyubid dynasty , the Mamluks , a Turco - Circassian military caste, took control about 1250. By the late 13th century, Egypt linked the Red Sea, India, Malaya, and East Indies. The mid-14th-century Black Death killed about 40% of

7140-859: The Anglo-Egyptian Condominium Agreement was signed: the Agreement stated that Sudan would be jointly governed by the Khedivate of Egypt and the United Kingdom. However, actual control of Sudan was in British hands only. In 1906, the Denshawai incident prompted many neutral Egyptians to join the nationalist movement. In 1914 the Ottoman Empire entered World War I in alliance with the Central Empires; Khedive Abbas II (who had grown increasingly hostile to

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7280-481: The Badarian by about seven hundred years. Contemporaneous Lower Egyptian communities coexisted with their southern counterparts for more than two thousand years, remaining culturally distinct, but maintaining frequent contact through trade. The earliest known evidence of Egyptian hieroglyphic inscriptions appeared during the predynastic period on Naqada III pottery vessels, dated to about 3200 BCE. A unified kingdom

7420-487: The British in preceding years) decided to support the motherland in war. Following such decision, the British forcibly removed him from power and replaced him with his brother Hussein Kamel . Hussein Kamel declared Egypt's independence from the Ottoman Empire, assuming the title of Sultan of Egypt . Shortly following independence, Egypt was declared a protectorate of the United Kingdom. After World War I , Saad Zaghlul and

7560-674: The Council of Regency, led by Prince Muhammad Abdel Moneim was formed. The council, however, held only nominal authority and the real power was actually in the hands of the Revolutionary Command Council , led by Naguib and Nasser. Popular expectations for immediate reforms led to the workers' riots in Kafr Dawar on 12 August 1952. Following a brief experiment with civilian rule, the Free Officers abrogated

7700-608: The Egyptian Auxiliary Police were observed helping the guerrillas. In response, on 25 January, General George Erskine sent out British tanks and infantry to surround the auxiliary police station in Ismailia. The police commander called the Interior Minister, Fouad Serageddin , Nahas's right-hand man, to ask if he should surrender or fight. Serageddin ordered the police to fight "to the last man and

7840-722: The Eighteenth Dynasty, marking the rise of Egypt as an international power that expanded during its greatest extension to an empire as far south as Tombos in Nubia , and included parts of the Levant in the east. This period is noted for some of the most well known Pharaohs , including Hatshepsut , Thutmose III , Akhenaten and his wife Nefertiti , Tutankhamun and Ramesses II . The first historically attested expression of monotheism came during this period as Atenism . Frequent contacts with other nations brought new ideas to

7980-686: The Evangelist in the 1st century. Diocletian 's reign (284–305 CE) marked the transition from the Roman to the Byzantine era in Egypt, when a great number of Egyptian Christians were persecuted. The New Testament had by then been translated into Egyptian. After the Council of Chalcedon in CE 451, a distinct Egyptian Coptic Church was firmly established. The Byzantines were able to regain control of

8120-759: The Italian army invaded Egypt. The Egyptian army did no fighting. In June 1940 the King dismissed Prime Minister Aly Maher, who got on poorly with the British. A new coalition Government was formed with the Independent Hassan Pasha Sabri as prime minister. Following a ministerial crisis in February 1942, the ambassador Sir Miles Lampson , pressed Farouk to have a Wafd or Wafd-coalition government replace Hussein Sirri Pasha 's government. On

8260-423: The Magi and the priests: and in addition to these he is related to have studied under the Brahmans (these are Indian philosophers); and from some he gathered astrology, from others geometry, and arithmetic and music from others, and different things from different nations, and only from the wise men of Greece did he get nothing, wedded as they were to a poverty and dearth of wisdom: so on the contrary he himself became

8400-429: The Muslim Brotherhood leaders and activists have either been sentenced to death or life imprisonment in a series of mass trials. On 18 January 2014, the interim government instituted a new constitution following a referendum approved by an overwhelming majority of voters (98.1%). 38.6% of registered voters participated in the referendum a higher number than the 33% who voted in a referendum during Morsi's tenure. In

8540-572: The Muslim Brotherhood. Liberal and secular groups walked out of the constituent assembly because they believed that it would impose strict Islamic practices, while Muslim Brotherhood backers threw their support behind Morsi. On 22 November 2012, President Morsi issued a temporary declaration immunising his decrees from challenge and seeking to protect the work of the constituent assembly. The move led to massive protests and violent action throughout Egypt. On 5 December 2012, tens of thousands of supporters and opponents of President Morsi clashed, in what

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8680-502: The New Kingdom. The country was later invaded and conquered by Libyans , Nubians and Assyrians , but native Egyptians eventually drove them out and regained control of their country. In 525 BCE, the Achaemenid Empire , led by Cambyses II , began their conquest of Egypt, eventually capturing the pharaoh Psamtik III at the battle of Pelusium . Cambyses II then assumed the formal title of pharaoh , but ruled Egypt from his home of Susa in Persia (modern Iran), leaving Egypt under

8820-453: The Nile River where they developed a settled agricultural economy and more centralised society. By about 6000 BCE, a Neolithic culture took root in the Nile Valley. During the Neolithic era, several predynastic cultures developed independently in Upper and Lower Egypt . The Badarian culture and the successor Naqada series are generally regarded as precursors to dynastic Egypt . The earliest known Lower Egyptian site, Merimda, predates

8960-503: The Security Police started rioting, protesting against reports that their term of duty was to be extended from 3 to 4 years. Hotels, nightclubs, restaurants and casinos were attacked in Cairo and there were riots in other cities. A day time curfew was imposed. It took the army 3 days to restore order. 107 people were killed. In the 1980s, 1990s, and 2000s, terrorist attacks in Egypt became numerous and severe, and began to target Christian Copts , foreign tourists and government officials. In

9100-442: The Shura Council and installed a temporary interim government. On 4 July 2013, 68-year-old Chief Justice of the Supreme Constitutional Court of Egypt Adly Mansour was sworn in as acting president over the new government following the removal of Morsi. The new Egyptian authorities cracked down on the Muslim Brotherhood and its supporters, jailing thousands and forcefully dispersing pro-Morsi and pro-Brotherhood protests. Many of

9240-418: The Sinai Peninsula border with Israel, and on 23 May he closed the Straits of Tiran to Israeli shipping. On 26 May Nasser declared, "The battle will be a general one and our basic objective will be to destroy Israel". This prompted the beginning of the Third Arab Israeli War (Six-Day War) in which Israel attacked Egypt, and occupied Sinai Peninsula and the Gaza Strip , which Egypt had occupied since

9380-811: The Soviet advisors in 1972. Egypt was renamed as Arab Republic of Egypt in 1971. Sadat launched the Infitah economic reform policy, while clamping down on religious and secular opposition. In 1973, Egypt, along with Syria, launched the Fourth Arab-Israeli War (Yom Kippur War), a surprise attack to regain part of the Sinai territory Israel had captured 6 years earlier. In 1975, Sadat shifted Nasser's economic policies and sought to use his popularity to reduce government regulations and encourage foreign investment through his programme of Infitah. Through this policy, incentives such as reduced taxes and import tariffs attracted some investors, but investments were mainly directed at low risk and profitable ventures like tourism and construction, abandoning Egypt's infant industries. Because of

9520-411: The Suez Canal on 26 July 1956; his hostile approach towards Israel and economic nationalism prompted the beginning of the Second Arab-Israeli War (Suez Crisis), in which Israel (with support from France and the United Kingdom) occupied the Sinai peninsula and the Canal. The war came to an end because of US and USSR diplomatic intervention and the status quo was restored. In 1958, Egypt and Syria formed

9660-417: The Suez Canal. As the British refused to leave their base around the Suez Canal, the Egyptian government cut off the water and refused to allow food into the Suez Canal base, announced a boycott of British goods, forbade Egyptian workers from entering the base and sponsored guerrilla attacks. On 24 January 1952, Egyptian guerrillas staged a fierce attack on the British forces around the Suez Canal, during which

9800-451: The Wafd for the Black Saturday riot, and dismissed Nahas as prime minister the next day. He was replaced by Aly Maher Pasha . On 22–23 July 1952, the Free Officers Movement , led by Muhammad Naguib and Gamal Abdel Nasser , launched a coup d'état ( Egyptian Revolution of 1952 ) against the king. Farouk I abdicated the throne to his son Fouad II , who was, at the time, a seven-month-old baby. The Royal Family left Egypt some days later and

9940-414: The War. Anti-monarchy sentiments further increased following the disastrous performance of the Kingdom in the First Arab-Israeli War . The 1950 election saw a landslide victory of the nationalist Wafd Party and the King was forced to appoint Mostafa El-Nahas as new prime minister. In 1951 Egypt unilaterally withdrew from the Anglo-Egyptian Treaty of 1936 and ordered all remaining British troops to leave

10080-424: The area. Over his lifetime, Fermat made the following contributions to the field: The interest of Leonhard Euler (1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateur Goldbach , pointed him towards some of Fermat's work on the subject. This has been called the "rebirth" of modern number theory, after Fermat's relative lack of success in getting his contemporaries' attention for

10220-542: The arrival of the first foreign ruling dynasty in Egypt, that of the Semitic Hyksos . The Hyksos invaders took over much of Lower Egypt around 1650 BCE and founded a new capital at Avaris . They were driven out by an Upper Egyptian force led by Ahmose I , who founded the Eighteenth Dynasty and relocated the capital from Memphis to Thebes . The New Kingdom c.  1550–1070 BCE began with

10360-668: The author of instruction to the Greeks in the learning which he had procured from abroad." Aristotle claimed that the philosophy of Plato closely followed the teachings of the Pythagoreans, and Cicero repeats this claim: Platonem ferunt didicisse Pythagorea omnia ("They say Plato learned all things Pythagorean"). Plato had a keen interest in mathematics, and distinguished clearly between arithmetic and calculation. (By arithmetic he meant, in part, theorising on number, rather than what arithmetic or number theory have come to mean.) It

10500-524: The battle of Heliopolis. Amr next proceeded in the direction of Alexandria, which surrendered to him by a treaty signed on 8 November 641. Alexandria was regained for the Byzantine Empire in 645 but was retaken by Amr in 646. In 654 an invasion fleet sent by Constans II was repulsed. The Arabs founded the capital of Egypt called Fustat , which was later burned down during the Crusades. Cairo

10640-530: The bulk of the swelling middle class in Egypt under Nasser. During the 1960s, the Egyptian economy went from sluggish to the verge of collapse, the society became less free, and Nasser's appeal waned considerably. In 1970, President Nasser died and was succeeded by Anwar Sadat . During his period , Sadat switched Egypt's Cold War allegiance from the Soviet Union to the United States, expelling

10780-454: The complex numbers C are an extension of the reals R , and the reals R are an extension of the rationals Q .) Classifying the possible extensions of a given number field is a difficult and partially open problem. Abelian extensions—that is, extensions L of K such that the Galois group Gal( L / K ) of L over K is an abelian group —are relatively well understood. Their classification

10920-534: The control of a satrapy . The entire Twenty-seventh Dynasty of Egypt , from 525 to 402 BCE, save for Petubastis III , was an entirely Achaemenid-ruled period, with the Achaemenid emperors all being granted the title of pharaoh. A few temporarily successful revolts against the Achaemenids marked the fifth century BCE, but Egypt was never able to permanently overthrow the Achaemenids. The Thirtieth Dynasty

11060-691: The country after a brief Sasanid Persian invasion early in the 7th century amidst the Byzantine–Sasanian War of 602–628 during which they established a new short-lived province for ten years known as Sasanian Egypt , until 639–42, when Egypt was invaded and conquered by the Islamic caliphate by the Muslim Arabs. When they defeated the Byzantine armies in Egypt, the Arabs brought Islam to

11200-505: The country's population. Egypt was conquered by the Ottoman Turks in 1517, after which it became a province of the Ottoman Empire . The defensive militarisation damaged its civil society and economic institutions. The weakening of the economic system combined with the effects of plague left Egypt vulnerable to foreign invasion. Portuguese traders took over their trade. Between 1687 and 1731, Egypt experienced six famines. The 1784 famine cost it roughly one-sixth of its population. Egypt

11340-399: The country. Some time during this period, Egyptians began to blend in their new faith with indigenous beliefs and practices, leading to various Sufi orders that have flourished to this day. These earlier rites had survived the period of Coptic Christianity . In 639 an army was sent in Egypt by the second caliph , Umar , under the command of Amr ibn al-As . They defeated a Roman army at

11480-465: The country: he built industries, a system of canals for irrigation and transport, and reformed the civil service. He constructed a military state with around four percent of the populace serving the army to raise Egypt to a powerful positioning in the Ottoman Empire in a way showing various similarities to the Soviet strategies (without communism) conducted in the 20th century. Muhammad Ali Pasha evolved

11620-509: The derived term [REDACTED] , Mu-ṣur . There is evidence of rock carvings along the Nile terraces and in desert oases. In the 10th millennium BCE , a culture of hunter-gatherers and fishers was replaced by a grain -grinding culture. Climate changes or overgrazing around 8000 BCE began to desiccate the pastoral lands of Egypt, forming the Sahara . Early tribal peoples migrated to

11760-419: The diagonal which has been subtracted such that the width..." The table's layout suggests that it was constructed by means of what amounts, in modern language, to the identity which is implicit in routine Old Babylonian exercises. If some other method was used, the triples were first constructed and then reordered by c / a {\displaystyle c/a} , presumably for actual use as

11900-503: The discussion of quadratic forms in Disquisitiones arithmeticae can be restated in terms of ideals and norms in quadratic fields. (A quadratic field consists of all numbers of the form a + b d {\displaystyle a+b{\sqrt {d}}} , where a {\displaystyle a} and b {\displaystyle b} are rational numbers and d {\displaystyle d}

12040-428: The dramatic growth in university education, and government support to national industries greatly improved social mobility and flattened the social curve. From academic year 1953–54 through 1965–66, overall public school enrolments more than doubled. Millions of previously poor Egyptians, through education and jobs in the public sector, joined the middle class. Doctors, engineers, teachers, lawyers, journalists, constituted

12180-475: The early Hellenistic period . In the case of number theory, this means, by and large, Plato and Euclid , respectively. While Asian mathematics influenced Greek and Hellenistic learning, it seems to be the case that Greek mathematics is also an indigenous tradition. Eusebius , PE X, chapter 4 mentions of Pythagoras : "In fact the said Pythagoras, while busily studying the wisdom of each nation, visited Babylon, and Egypt, and all Persia, being instructed by

12320-580: The election of Nasser in 1956. In October 1954, Egypt and the United Kingdom agreed to abolish the Anglo-Egyptian Condominium Agreement of 1899 and grant Sudan independence; the agreement came into force on 1 January 1956. Nasser assumed power as president in June 1956 and began dominating the history of modern Egypt . British forces completed their withdrawal from the occupied Suez Canal Zone on 13 June 1956. He nationalised

12460-458: The elimination of subsidies on basic foodstuffs, it led to the 1977 Egyptian Bread Riots . Sadat made a historic visit to Israel in 1977 , which led to the 1979 Egypt-Israel peace treaty in exchange for Israeli withdrawal from Sinai. In return, Egypt recognised Israel as a legitimate sovereign state. Sadat's initiative sparked enormous controversy in the Arab world and led to Egypt's expulsion from

12600-462: The finite generation of the group of rational points on an elliptic curve is in Chapter 16, and integer points on the Mordell curve in Chapter 26. In a hostile review of Lang's book, Mordell wrote: In recent times, powerful new geometric ideas and methods have been developed by means of which important new arithmetical theorems and related results have been found and proved and some of these are not easily proved otherwise. Further, there has been

12740-469: The first President of the Republic, serving in that capacity for a little under one and a half years. Republic of Egypt (1953–1958) was declared. Naguib was forced to resign in 1954 by Gamal Abdel Nasser  – a Pan-Arabist and the real architect of the 1952 movement – and was later put under house arrest . After Naguib's resignation, the position of president was vacant until

12880-456: The formation of the first nationalist groupings in 1879, with Ahmed ʻUrabi a prominent figure. After increasing tensions and nationalist revolts, the United Kingdom invaded Egypt in 1882, crushing the Egyptian army at the Battle of Tell El Kebir and militarily occupying the country. Following this, the Khedivate became a de facto British protectorate under nominal Ottoman sovereignty. In 1899

13020-404: The geometric approach, the consideration of homogeneous equations and homogeneous co-ordinates is fundamental, for the same reasons that projective geometry is the dominant approach in algebraic geometry. Rational number solutions therefore are the primary consideration; but integral solutions (i.e. lattice points ) can be treated in the same way as an affine variety may be considered inside

13160-508: The greatest common divisor of two numbers (the Euclidean algorithm ; Elements , Prop. VII.2) and the first known proof of the infinitude of primes ( Elements , Prop. IX.20). In 1773, Lessing published an epigram he had found in a manuscript during his work as a librarian; it claimed to be a letter sent by Archimedes to Eratosthenes . The epigram proposed what has become known as Archimedes's cattle problem ; its solution (absent from

13300-633: The group depended for support. During Mubarak's reign, the political scene was dominated by the National Democratic Party , which was created by Sadat in 1978. It passed the 1993 Syndicates Law, 1995 Press Law, and 1999 Nongovernmental Associations Law which hampered freedoms of association and expression by imposing new regulations and draconian penalties on violations. As a result, by the late 1990s parliamentary politics had become virtually irrelevant and alternative avenues for political expression were curtailed as well. Cairo grew into

13440-559: The imposition of British and French controllers who sat in the Egyptian cabinet, and, "with the financial power of the bondholders behind them, were the real power in the Government." Other circumstances like epidemic diseases (cattle disease in the 1880s), floods and wars drove the economic downturn and increased Egypt's dependency on foreign debt even further. Local dissatisfaction with the Khedive and with European intrusion led to

13580-873: The lack of unique factorisation in algebraic number fields. (For example, in the field generated by the rationals and − 5 {\displaystyle {\sqrt {-5}}} , the number 6 {\displaystyle 6} can be factorised both as 6 = 2 ⋅ 3 {\displaystyle 6=2\cdot 3} and 6 = ( 1 + − 5 ) ( 1 − − 5 ) {\displaystyle 6=(1+{\sqrt {-5}})(1-{\sqrt {-5}})} ; all of 2 {\displaystyle 2} , 3 {\displaystyle 3} , 1 + − 5 {\displaystyle 1+{\sqrt {-5}}} and 1 − − 5 {\displaystyle 1-{\sqrt {-5}}} are irreducible, and thus, in

13720-524: The last bullet". The resulting battle saw the police station levelled and 43 Egyptian policemen killed together with 3 British soldiers. The Ismailia incident outraged Egypt. The next day, 26 January 1952 was "Black Saturday" , as the anti-British riot was known, that saw much of downtown Cairo which the Khedive Ismail the Magnificent had rebuilt in the style of Paris, burned down. Farouk blamed

13860-472: The latter ( Diophantine approximation ). The older term for number theory is arithmetic . By the early twentieth century, it had been superseded by number theory . (The word arithmetic is used by the general public to mean " elementary calculations "; it has also acquired other meanings in mathematical logic , as in Peano arithmetic , and computer science , as in floating-point arithmetic .) The use of

14000-403: The lines later developed fully by Gauss. In his old age, he was the first to prove Fermat's Last Theorem for n = 5 {\displaystyle n=5} (completing work by Peter Gustav Lejeune Dirichlet , and crediting both him and Sophie Germain ). In his Disquisitiones Arithmeticae (1798), Carl Friedrich Gauss (1777–1855) proved the law of quadratic reciprocity and developed

14140-509: The manuscript) requires solving an indeterminate quadratic equation (which reduces to what would later be misnamed Pell's equation ). As far as it is known, such equations were first successfully treated by the Indian school . It is not known whether Archimedes himself had a method of solution. Very little is known about Diophantus of Alexandria ; he probably lived in the third century AD, that is, about five hundred years after Euclid. Six out of

14280-455: The military from one that convened under the tradition of the corvée to a great modernised army. He introduced conscription of the male peasantry in 19th century Egypt, and took a novel approach to create his great army, strengthening it with numbers and in skill. Education and training of the new soldiers became mandatory; the new concepts were furthermore enforced by isolation. The men were held in barracks to avoid distraction of their growth as

14420-554: The monarchy and the 1923 constitution and declared Egypt a republic on 18 June 1953. Naguib was proclaimed as president, while Nasser was appointed as the new prime minister. Following the 1952 Revolution by the Free Officers Movement , the rule of Egypt passed to military hands and all political parties were banned. On 18 June 1953, the Egyptian Republic was declared, with General Muhammad Naguib as

14560-425: The most important tools of analytic number theory are the circle method , sieve methods and L-functions (or, rather, the study of their properties). The theory of modular forms (and, more generally, automorphic forms ) also occupies an increasingly central place in the toolbox of analytic number theory. One may ask analytic questions about algebraic numbers , and use analytic means to answer such questions; it

14700-693: The most interesting questions in each area remain open and are being actively worked on. The term elementary generally denotes a method that does not use complex analysis . For example, the prime number theorem was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by Erdős and Selberg . The term is somewhat ambiguous: for example, proofs based on complex Tauberian theorems (for example, Wiener–Ikehara ) are often seen as quite enlightening but not elementary, in spite of using Fourier analysis , rather than complex analysis as such. Here as elsewhere, an elementary proof may be longer and more difficult for most readers than

14840-606: The news. The Egyptian military then assumed the power to govern. Mohamed Hussein Tantawi , chairman of the Supreme Council of the Armed Forces , became the de facto interim head of state . On 13 February 2011, the military dissolved the parliament and suspended the constitution. A constitutional referendum was held on 19 March 2011. On 28 November 2011, Egypt held its first parliamentary election since

14980-460: The night of 4 February 1942, British troops and tanks surrounded Abdeen Palace in Cairo and Lampson presented Farouk with an ultimatum . Farouk capitulated, and Nahhas formed a government shortly thereafter. Most British troops were withdrawn to the Suez Canal area in 1947 (although the British army maintained a military base in the area), but nationalist, anti-British feelings continued to grow after

15120-435: The one hand, and lengths and proportions (which may be identified with real numbers, whether rational or not), on the other hand. The Pythagorean tradition spoke also of so-called polygonal or figurate numbers . While square numbers , cubic numbers , etc., are seen now as more natural than triangular numbers , pentagonal numbers , etc., the study of the sums of triangular and pentagonal numbers would prove fruitful in

15260-625: The president power to dissolve parliament and end judicial election monitoring. In 2009, Dr. Ali El Deen Hilal Dessouki, Media Secretary of the National Democratic Party ( NDP ), described Egypt as a " pharaonic " political system, and democracy as a "long-term goal". Dessouki also stated that "the real center of power in Egypt is the military". On 25 January 2011, widespread protests began against Mubarak's government. On 11 February 2011, Mubarak resigned and fled Cairo. Jubilant celebrations broke out in Cairo's Tahrir Square at

15400-499: The previous regime had been in power. Turnout was high and there were no reports of major irregularities or violence. Mohamed Morsi , who was affiliated with the Muslim Brotherhood , was elected president on 24 June 2012. On 30 June 2012, Mohamed Morsi was sworn in as Egypt's president. On 2 August 2012, Egypt's Prime Minister Hisham Qandil announced his 35-member cabinet comprising 28 newcomers, including four from

15540-447: The proof "is of a very advanced character" (p. 263). Despite a bad press initially, Lang's conception has been sufficiently widely accepted for a 2006 tribute to call the book "visionary". A larger field sometimes called arithmetic of abelian varieties now includes Diophantine geometry along with class field theory , complex multiplication , local zeta-functions and L-functions . Paul Vojta wrote: A single equation defines

15680-537: The provincial capital of the Mamluk Sultanate in the 13th century. Egypt then became part of the Ottoman Empire in 1517, before its local ruler Muhammad Ali established modern Egypt as an autonomous Khedivate in 1867. The country was then occupied by the British Empire along with Sudan and gained independence in 1922 as a monarchy . Following the 1952 revolution , Egypt declared itself

15820-592: The region, especially the battles in North Africa against Italy and Germany. Its highest priorities were control of the Eastern Mediterranean, and especially keeping the Suez Canal open for merchant ships and for military connections with India and Australia. When the war began in September 1939, Egypt declared martial law and broke off diplomatic relations with Germany. It broke diplomatic relations with Italy in 1940, but never declared war, even when

15960-753: The roots of the subject. This is an example of a general procedure in analytic number theory: deriving information about the distribution of a sequence (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complex-valued function. An algebraic number is any complex number that is a solution to some polynomial equation f ( x ) = 0 {\displaystyle f(x)=0} with rational coefficients; for example, every solution x {\displaystyle x} of x 5 + ( 11 / 2 ) x 3 − 7 x 2 + 9 = 0 {\displaystyle x^{5}+(11/2)x^{3}-7x^{2}+9=0} (say)

16100-567: The second rather than the first definition: some of sieve theory, for instance, uses little analysis, yet it does belong to analytic number theory. The following are examples of problems in analytic number theory: the prime number theorem , the Goldbach conjecture (or the twin prime conjecture , or the Hardy–Littlewood conjectures ), the Waring problem and the Riemann hypothesis . Some of

16240-711: The subject. Euler's work on number theory includes the following: Joseph-Louis Lagrange (1736–1813) was the first to give full proofs of some of Fermat's and Euler's work and observations—for instance, the four-square theorem and the basic theory of the misnamed "Pell's equation" (for which an algorithmic solution was found by Fermat and his contemporaries, and also by Jayadeva and Bhaskara II before them.) He also studied quadratic forms in full generality (as opposed to m X 2 + n Y 2 {\displaystyle mX^{2}+nY^{2}} )—defining their equivalence relation, showing how to put them in reduced form, etc. Adrien-Marie Legendre (1752–1833)

16380-456: The systematic study of indefinite quadratic equations—in particular, the misnamed Pell equation , in which Archimedes may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler. Later Sanskrit authors would follow, using Brahmagupta's technical terminology. A general procedure (the chakravala , or "cyclic method") for solving Pell's equation

16520-401: The term arithmetic for number theory regained some ground in the second half of the 20th century, arguably in part due to French influence. In particular, arithmetical is commonly preferred as an adjective to number-theoretic . The earliest historical find of an arithmetical nature is a fragment of a table: the broken clay tablet Plimpton 322 ( Larsa, Mesopotamia , ca. 1800 BC) contains

16660-516: The theory of quadratic forms (in particular, defining their composition). He also introduced some basic notation ( congruences ) and devoted a section to computational matters, including primality tests. The last section of the Disquisitiones established a link between roots of unity and number theory: The theory of the division of the circle...which is treated in sec. 7 does not belong by itself to arithmetic, but its principles can only be drawn from higher arithmetic. In this way, Gauss arguably made

16800-540: The thirteen books of Diophantus's Arithmetica survive in the original Greek and four more survive in an Arabic translation. The Arithmetica is a collection of worked-out problems where the task is invariably to find rational solutions to a system of polynomial equations, usually of the form f ( x , y ) = z 2 {\displaystyle f(x,y)=z^{2}} or f ( x , y , z ) = w 2 {\displaystyle f(x,y,z)=w^{2}} . Thus, nowadays,

16940-532: The title of King of Egypt ; despite being nominally independent, the Kingdom was still under British military occupation and the UK still had great influence over the state. The new government drafted and implemented a constitution in 1923 based on a parliamentary system. The nationalist Wafd Party won a landslide victory in the 1923–1924 election and Saad Zaghloul was appointed as the new prime minister. In 1936,

17080-571: The war sank into a stalemate. In mid May 1967, the Soviet Union issued warnings to Nasser of an impending Israeli attack on Syria. Although the chief of staff Mohamed Fawzi verified them as "baseless", Nasser took three successive steps that made the war virtually inevitable: on 14 May he deployed his troops in Sinai near the border with Israel, on 19 May he expelled the UN peacekeepers stationed in

17220-421: The work of Bernhard Riemann (1859) on the zeta function is the canonical starting point; Jacobi's four-square theorem (1839), which predates it, belongs to an initially different strand that has by now taken a leading role in analytic number theory ( modular forms ). The history of each subfield is briefly addressed in its own section below; see the main article of each subfield for fuller treatments. Many of

17360-473: Was also allowed to run for a third term in the next election in 2024. Under El-Sisi Egypt is said to have returned to authoritarianism. New constitutional reforms have been implemented, meaning strengthening the role of military and limiting the political opposition. The constitutional changes were accepted in a referendum in April 2019. In December 2020, final results of the parliamentary election confirmed

17500-542: Was always a difficult province for the Ottoman Sultans to control, due in part to the continuing power and influence of the Mamluks , the Egyptian military caste who had ruled the country for centuries. Egypt remained semi-autonomous under the Mamluks until it was invaded by the French forces of Napoleon Bonaparte in 1798. After the French were defeated by the British, a three-way power struggle ensued between

17640-537: Was borrowed into Coptic as " gyptios ", and from there into Arabic as " qubṭī ", back formed into " قبط " (" qubṭ "), whence English " Copt ". Prominent Ancient Greek historian and Geographer , Strabo , provided a folk etymology stating that " Αἴγυπτος " (Aigýptios) had originally evolved as a compound from " Aἰγαίου ὑπτίως " Aegaeou huptiōs , meaning " Below the Aegean ". " Miṣr " ( Arabic pronunciation: [misˤɾ] ; " مِصر ")

17780-476: Was described as the largest violent battle between Islamists and their foes since the country's revolution. Mohamed Morsi offered a "national dialogue" with opposition leaders but refused to cancel the December 2012 constitutional referendum . On 3 July 2013, after a wave of public discontent with autocratic excesses of Morsi's Muslim Brotherhood government, the military removed Morsi from office, dissolved

17920-648: Was done in western Europe during the Middle Ages. Matters started to change in Europe in the late Renaissance , thanks to a renewed study of the works of Greek antiquity. A catalyst was the textual emendation and translation into Latin of Diophantus' Arithmetica . Pierre de Fermat (1607–1665) never published his writings; in particular, his work on number theory is contained almost entirely in letters to mathematicians and in private marginal notes. In his notes and letters, he scarcely wrote any proofs—he had no models in

18060-513: Was elected in 2014, has been described by a number of watchdogs as authoritarian and responsible for perpetuating the country's poor human rights record . Islam is the official religion and Arabic is official language. Egypt is considered to be a regional power in the Middle East, North Africa and the Muslim world , and a middle power worldwide. It is a developing country . Egypt is

18200-543: Was exceptionally well developed and included the foundations of modern elementary algebra . Late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt . In book nine of Euclid's Elements , propositions 21–34 are very probably influenced by Pythagorean teachings ; it is very simple material ("odd times even

18340-717: Was finally found by Jayadeva (cited in the eleventh century; his work is otherwise lost); the earliest surviving exposition appears in Bhāskara II 's Bīja-gaṇita (twelfth century). Indian mathematics remained largely unknown in Europe until the late eighteenth century; Brahmagupta and Bhāskara's work was translated into English in 1817 by Henry Colebrooke . In the early ninth century, the caliph Al-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work (the Sindhind , which may or may not be Brahmagupta's Brāhmasphuṭasiddhānta ). Diophantus's main work,

18480-508: Was founded c.  3150 BCE by King Menes , leading to a series of dynasties that ruled Egypt for the next three millennia. Egyptian culture flourished during this long period and remained distinctively Egyptian in its religion , arts , language and customs. The first two ruling dynasties of a unified Egypt set the stage for the Old Kingdom period, c.  2700–2200 BCE, which constructed many pyramids , most notably

18620-443: Was granted the status of an autonomous vassal state or Khedivate (1867–1914) in 1867. The Suez Canal , built in partnership with the French, was completed in 1869. Its construction was financed by European banks. Large sums also went to patronage and corruption. New taxes caused popular discontent. In 1875 Isma'il avoided bankruptcy by selling all Egypt's shares in the canal to the British government. Within three years this led to

18760-414: Was later built in the year 986 to grow to become the largest and richest city in the Arab caliphate , second only to Baghdad . The Abbasid period was marked by new taxations, and the Copts revolted again in the fourth year of Abbasid rule. At the beginning of the 9th century the practice of ruling Egypt through a governor was resumed under Abdallah ibn Tahir , who decided to reside at Baghdad , sending

18900-424: Was less than 25%. Election observers also alleged government interference in the election process. After the election, Mubarak imprisoned Ayman Nour , the runner-up. Human Rights Watch's 2006 report on Egypt detailed serious human rights violations under Mubarak's rule, including routine torture , arbitrary detentions and trials before military and state security courts. In 2007, Amnesty International released

19040-401: Was the first to state the law of quadratic reciprocity. He also conjectured what amounts to the prime number theorem and Dirichlet's theorem on arithmetic progressions . He gave a full treatment of the equation a x 2 + b y 2 + c z 2 = 0 {\displaystyle ax^{2}+by^{2}+cz^{2}=0} and worked on quadratic forms along

19180-509: Was the last native ruling dynasty during the Pharaonic epoch. It fell to the Achaemenids again in 343 BCE after the last native Pharaoh, King Nectanebo II , was defeated in battle. This Thirty-first Dynasty of Egypt , however, did not last long, as the Achaemenids were toppled several decades later by Alexander the Great . The Macedonian Greek general of Alexander, Ptolemy I Soter , founded

19320-524: Was the object of the programme of class field theory , which was initiated in the late 19th century (partly by Kronecker and Eisenstein ) and carried out largely in 1900–1950. An example of an active area of research in algebraic number theory is Iwasawa theory . The Langlands program , one of the main current large-scale research plans in mathematics, is sometimes described as an attempt to generalise class field theory to non-abelian extensions of number fields. The central problem of Diophantine geometry

19460-400: Was thus arguably a pioneer in the study of number systems . (Book X of Euclid's Elements is described by Pappus as being largely based on Theaetetus's work.) Euclid devoted part of his Elements to prime numbers and divisibility, topics that belong unambiguously to number theory and are basic to it (Books VII to IX of Euclid's Elements ). In particular, he gave an algorithm for computing

19600-456: Was translated into English in early 19th century by British missionary Alexander Wylie . There is also some numerical mysticism in Chinese mathematics, but, unlike that of the Pythagoreans, it seems to have led nowhere. Aside from a few fragments, the mathematics of Classical Greece is known to us either through the reports of contemporary non-mathematicians or through mathematical works from

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