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Grande Odalisque

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Grande Odalisque
Jean Auguste Dominique Ingres, La Grande Odalisque, 1814.jpg
Artist Jean Auguste Dominique Ingres
Year 1814
Type Oil on canvas
Dimensions 88.9 cm × 162.56 cm (35 in × 64 in)
Location Louvre, Paris

Grande Odalisque, also known as Une Odalisque or La Grande Odalisque, is an oil painting of 1814 by Jean Auguste Dominique Ingres depicting an odalisque, or concubine. Ingres' contemporaries considered the work to signify Ingres' break from Neoclassicism, indicating a shift toward exotic Romanticism.

Grande Odalisque attracted wide criticism when it was first shown. It has been especially noted for the elongated proportions and lack of anatomical realism. The work is displayed in the Louvre, Paris.

The painting was commissioned by Napoleon's sister, Queen Caroline Murat of Naples,[1] and finished in 1814. Ingres drew upon works such as Dresden Venus by Giorgione, and Titian's Venus of Urbino as inspiration for his reclining nude figure, though the actual pose of a reclining figure looking back over her shoulder is directly drawn from the 1809 Portrait of Madame Récamier by Jacques-Louis David.

Madame Récamier painted by Jacques-Louis David (1800).

Ingres portrays a concubine in languid pose as seen from behind with distorted proportions. The small head, elongated limbs, and cool color scheme all reveal influences from Mannerists such as Parmigianino,[2] whose Madonna with the Long Neck was also famous for anatomical distortion.

This eclectic mix of styles, combining classical form with Romantic themes, prompted harsh criticism when it was first shown in 1814. Critics viewed Ingres as a rebel against the contemporary style of form and content. When the painting was first shown in the Salon of 1819, one critic remarked that the work had "neither bones nor muscle, neither blood, nor life, nor relief, indeed nothing that constitutes imitation".[3] This echoed the general view that Ingres had disregarded anatomical realism.[4] Ingres instead favored long lines to convey curvature and sensuality, as well as abundant, even light to tone down the volume.[4] Ingres continued to be criticized for his work until the mid-1820s.[2]

Stemming from the initial criticism the painting received, the figure in Grande Odalisque is thought to be drawn with "two or three vertebrae too many."[1][5] Critics at the time believed the elongations to be errors on the part of Ingres, but recent studies show the elongations to have been deliberate distortions.[6] Measurements taken on the proportions of real women showed that Ingres's figure was drawn with a curvature of the spine and rotation of the pelvis impossible to replicate.[5] It also showed the left arm of the odalisque is shorter than the right. The study concluded that the figure was longer by five instead of two or three vertebrae and that the excess affected the lengths of the pelvis and lower back instead of merely the lumbar region.[5]

Another interpretation of this painting suggests that since the duty of some concubines was merely to satisfy the carnal pleasures of the sultan, this elongation of her pelvic area may have been a symbolic distortion by Ingres. While this may represent sensuous feminine beauty, her gaze, on the other hand, has been said to "[reflect] a complex psychological make-up" or "[betray] no feeling". In addition, the distance between her gaze and her pelvic region may be a physical representation of the depth of thought and complex emotions of a woman's thoughts and feelings.[5]

In other works[edit]

La Grande Odalisque was appropriated by the feminist art group Guerrilla Girls for their first color poster and most iconic image. The 1989 Metropolitan Museum poster gave Ingres's odalisque a gorilla mask and posed the question "Do women have to be naked to get into the Met. Museum?". The poster used data from the group's first "weenie count" and drew attention to the overwhelming amount of female nudes counted in the Modern Art sections of The Met. The poster was rejected by the Public Art Fund in New York and was run in advertising space on New York City buses until the bus company cancelled the lease arguing that the image was "too suggestive and that the figure appeared to have more than a fan in her hand."[7][8]

  1. ^ a b Weston, Helen (1996). "A Look Back on Ingres". Oxford Art Journal 19 (2): 114–116. doi:10.1093/oaj/19.2.114.
  2. ^ a b Kleiner, Fred; Christian J. Mamiya (2005). Gardner’s Art Through the Ages (12 ed.). California: Wadsworth/Thompson Learning. pp. 826–827. ISBN 0-534-64091-5.
  3. ^ Benjamin, Roger (December 2000). "Ingres Chez Les Fauves". Art History 23 (5): 754–755. doi:10.1111/1467-8365.00242.
  4. ^ a b "Une Odalisque". Louvre Museum. Retrieved 2008-01-09.
  5. ^ a b c d Maigne, Jean-Yves; Gilles Chatellier; Hélène Norlöff (July 2004). "Extra vertebrae in Ingres' La Grande Odalisque". Journal of the Royal Society of Medicine 97 (7): 342–344. doi:10.1258/jrsm.97.7.342. PMC 1079534. PMID 15229267. Retrieved 2008-01-09.
  6. ^ Hautefeuille, Annie (2 July 2004). "Little extra out the back". The Australian. p. 16.
  7. ^ Manchester, Elizabeth (December 2004). "Guerrilla Girls [no title] 1985–90". Tate.
  8. ^ Chadwick, Whitney; Guerrilla Girls (1995). Confessions of the Guerrilla Girls. New York. p. 61.

External links[edit]

External video
Ingres' La Grand Odalisque, Smarthistory

Media related to La Grande Odalisque at Wikimedia Commons


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Mingei (民芸 lit. "folk arts" or "arts of the people"?), the Japanese folk art movement, was developed in the late 1920s and 1930s in Japan. Its founding father was Yanagi Sōetsu (1889–1961).

In 1916, Yanagi made his first trip to Korea out of a curiosity for Korean crafts. The trip led to the establishment of the Korean Folk Crafts Museum in 1924, and the coining of the term mingei by Yanagi, potters Hamada Shōji (1894–1978) and Kawai Kanjirō (1890–1966). In 1926, the Folk Art Movement was formally declared by Yanagi Sōetsu. Yanagi rescued lowly pots used by commoners in the Edo and Meiji period that were disappearing in rapidly urbanizing Japan. In 1936, the Japanese Folk Crafts Museum was established.

Mingei Theory[edit]

The philosophical pillar of mingei is “hand-crafted art of ordinary people” (民衆的な工芸 (minshū-teki-na kōgei?)). Yanagi Sōetsu discovered beauty in everyday ordinary and utilitarian objects created by nameless and unknown craftsmen. According to Yanagi, utilitarian objects made by the common people are “beyond beauty and ugliness”. Below are a few criteria of mingei art and crafts:

  • made by anonymous crafts people
  • produced by hand in quantity
  • inexpensive
  • used by the masses
  • functional in daily life
  • representative of the regions in which they were produced.

Yanagi's book The Unknown Craftsman has become an influential work since its first release in English in 1972. Yanagi's book examines the Japanese way of viewing and appreciating art and beauty in everyday crafts, including ceramics, lacquer, textiles, and woodwork.

Criticisms of Mingei Theory and Orientalism[edit]

In recent decades, scholars such as Yuko Kikuchi and Brian Moeran have claimed they have uncovered power relations and ultra-nationalism that lie at the core of the formation of mingei theory. In 1927, Yanagi put forward the "criterion of beauty in Japan" (日本における美の標準 (nihon ni okeru bi no hyōjun?)) in The Way of Crafts (工芸の道 (kōgei no michi?)). During the years of rising militarism in Japan, Sōetsu Yanagi extended his application of the "criterion of beauty" to the crafts of the Okinawans and the Ainu in the Japanese peripheries, and to those of the colonies including Korea, Taiwan and Manchuria. These scholars argue that mingei theory, far from being an Oriental theory, is a “hybridization” and “appropriation” of Occidental ideas such as that of William Morris (1834–1896)[1] which Bernard Leach (1887–1979) introduced when he lived and worked in Japan from 1909 to 1920.[2] Whereas Leach helped Japanese artists to rediscover their Oriental cultural origins in Occidental eyes, Japan applied Orientalism to its own art and projected the same Orientalism to the art of other Far Eastern countries such as Korea. Yuko Kikuchi terms this phenomenon “Oriental Orientalism”.

Yanagi’s “Korea and its Art” was severely criticized by Korean intellectuals as a “colonialist view of history” in 1974. Yanagi defined “beauty of sadness” (悲哀の美 (hiai no bi?)) as the “innate, original beauty created by the Korean race” (民族の固有の美 (minzoku no koyū no bi?)). Yanagi believed that the long history of foreign invasions of Korea was reflected in Korean art, and especially in its pottery, in the “sad and lonely” lines. Such a theory has been criticized by Korean scholars as the “aesthetic of colonialism”.

See also[edit]

References and further reading[edit]

  • Brandt, Kim. Kingdom of Beauty: Mingei and the Politics of Folk Art in Imperial Japan. Durham and London: Duke UP, 2007.
  • Kikuchi, Yuko. Japanese Modernization and Mingei Theory: Cultural Nationalism and Oriental Orientalism. London and New York: RoutledgeCurzon, 2004.
  • Saint-Gilles, Amaury. Mingei: Japan's Enduring Folk Arts. Union City, California: Heian International, 1983.
  • Yanagi, Soetsu. The Unknown Craftsman: A Japanese Insight into Beauty. Tokyo, New York: Kodansha International, 1989.
  • De Waal, Edmund. “Homo Orientals: Bernard Leach and the Image of the Japanese Craftsman.” Journal of Design History, Vol. 10, No. 4, Craft, Culture and Identity (1997): 355-362.
  • Karatani, Kojin and Kohso, Sabu. “Uses of Aesthetics: After Orientalism.” Boundary 2, Vol. 25, No. 2, Edward Said (Summer, 1998): 145-160.
  • Moeran, Brian. “Bernard Leach and the Japanese Folk Craft Movement: The Formative Years.” Journal of Design History, Vol. 2, No. 2/3, (1989): 139-144.

External links[edit]

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Murder of Ofir Rahum

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Murder of Ofir Rahum
Part of the Second Intifada militancy campaign
Israel outline jerusalem.png
Red pog.svg
The attack site
Location Outskirts of Ramallah, West Bank
Date 17 January 2001
Attack type
Shooting attack
Deaths a 16-year-old Israeli civilian (Ofir Rahum)
Perpetrator Three Palestinian gunmen (Mona Jaud Awana, Hassan Alkadi, and Abdul Fatah Doleh)

The Murder of Ofir Rahum was a shooting attack which occurred on 17 January 2001, in which Palestinian terrorists from the Tanzim faction of Fatah killed 16-year-old Israeli high school student Ofir Rahum on the outskirts of Ramallah.

The murder was planned and initiated by a Palestine Liberation Organization collaborator, 24-year-old Mona Jaud Awana from Bir Nabala. After many weeks in which Mona conducted long private conversations in English with Rahum through the instant messenger program ICQ, during which she masqueraded as a Jewish immigrant from Morocco named Sally, Mona managed to gain his confidence and get him to meet with her in Jerusalem, supposedly for romantic purposes. When he arrived for the meeting, she drove him through the border control into the Palestinian-controlled territory and towards a secluded area on the outskirts of Ramallah where Palestinian terrorists shot him at close range, with Mona standing aside and watching.[1][2]

Ofir Rahum

Mona Awana, later arrested by the Israeli police, stated that on the day of the lynching of two Israeli soldiers in Ramallah in late 2000 by a Palestinian mob, she decided to abduct an Israeli and murder him. Mona had been present at the Ramallah lynching, and said she was "excited" by what she saw. Soon after, she began contacting Israelis on the Internet.[3] Awana chatted with several Israeli teenagers on chat rooms and engaged in an online romance with Rahum. In conversations over several months she pressed for a meeting in Jerusalem. When Rahum suggested a venue closer to his home, she claimed she had no car but promised to arrange for his return by 5 P.M. Employing sexual innuendos, she persuaded him to meet her. Two days before the murder, she wrote, "You don't know how much I am waiting for Wednesday."[4] When they met, she convinced him to accompany her to Ramallah.[5]

On 17 January 2001, Mona and Rahum met at the central bus station in Jerusalem, spoke briefly in English, and then got on a taxi which drove to the el-Ram intersection. From there they drove in a Ford Escort with Israeli license plates, through the border control into the Palestinian-controlled territory[6] and towards the outskirts of Ramallah.[7] When Rahum questioned Mona why they were traveling through Arab-dominant areas, Mona replied that "that's the way it is in Jerusalem, which has both Arab and Jewish neighborhoods."[8]

At 11:30, Mona stopped the car at a secluded area on the outskirts of Ramallah next to a car with Palestinian license plates, at a location which was chosen in advance with the Palestinian militant cell. Mona immediately jumped out of the vehicle,[1] while the militants, Hassan Alkadi and Abdul Fatah Doleh, armed with rifles, got out of their car and shot Rahum from close range more than 15 times, while all along Mona was standing aside and watching. Afterwards, the assailants threw Rahum's body into the trunk and fled to Ramallah.[9]

The assailants secretly buried Rahum's body in the al-Tira neighborhood in Ramallah.[8]

After Rahum did not return home from school, his family began looking for him, and found out that he did not go to school that day. Following further investigation by the Israeli Shin Bet security agency and the Israeli International Crimes Unit, it was revealed that Rahum corresponded with Mona through the Internet and that she pressed Rahum to meet with her in Jerusalem, and that she promised to have intimate relations with him.[10][11]

Meanwhile, an undercover Israeli Duvdevan Unit force, in conjunction with the Shin Bet, raided Mona's house in Bir Nabala and managed to apprehend Mona.[8][12] She was later interrogated by the Shin Bet.[13]

On 18 January, Palestinian security forces located Rahum's body and brought it to the Ramallah hospital. A short time later, his body was transferred to Israel.

Although Mona initially refused to cooperate with the investigators, she broke down after about four weeks and gave a full confession of her involvement in the murder,[14] and revealed the identities of her partners in crime – the senior Ramallah-area Fatah Tanzim operatives Hassan Alkadi and Abdul Fatah Doleh of Beitunia.[8][15]

The murder caused great shock amongst the Israeli public. The Israeli Prime Minister and Defense Minister Ehud Barak stated in response to the murder that the attack was a "heinous crime committed by vile killers, lacking human dignity. The Security forces will not spare any effort to reach the murderers so that they would be punished severely."[16] Thousands[citation needed] of people attended Rahum's funeral which was held on 19 January at noon in the Ashkelon cemetery.[17] Rahum's gravestone was built in the form of a personal computer.[18]

Israel's Permanent Representative to the United Nations wrote to the Secretary-General to protest the "recent acts of Palestinian terrorism directed against Israeli civilians".[19]

Awana was tried, and eventually convicted of murder in February 2003[20] and sentenced to life imprisonment.[21] She was later transferred to the Neveh Tirza prison in 2006 after attacking another inmate.[22]

Though the Israeli government initially said that Awana would not be part of the deal,[7] she was released to Gaza on 18 October 2011 as part of the Gilad Shalit prisoner exchange between Israel and Hamas.[23] Awana is now residing in exile in Turkey.[24]

Abbas-Awana meeting[edit]

On 20 December 2011, Palestinian leader Mahmoud Abbas privately met Awana in Ankara, for which he was criticized by Israel. Abbas said that the release of the remaining Palestinian prisoners in Israel was at the top of the Palestinian agenda. An Israeli government spokesman said that the Israeli government was "disappointed" by this decision, and that "instead of promoting peace and reconciliation, the Palestinian leadership seems to be putting extremist murderers up on a pedestal. This raises serious questions as to their commitment and as to their desire to end the conflict."[25][26]

See also[edit]

  1. ^ a b Hershman, Tania (19 January 2001). "Israel's 'First Internet Murder'". Wired.
  2. ^ The Murder of Ofir Rahum
  3. ^
  4. ^ "Ofir's Fatal Attraction: An Israeli boy looks for love online and meets a vengeful Palestinian". Newsweek. 2 April 2001. Retrieved 29 September 2012.
  5. ^ Weimann, Gabriel (2006). Terror on the Internet: the new arena, the new challenges. US Institute of Peace Press. pp. 71–74. ISBN 1929223714.
  6. ^ Philps, Alan (20 January 2001). "Peace hope buried with shot boy". The Telegraph. Retrieved 29 September 2012.
  7. ^ a b Keinon, Herb (14 October 2011). "The reasons for Hamas's 'flexibility' on Schalit swap". The Jerusalem Post. Retrieved 31 May 2013.
  8. ^ a b c d "פיתיון האינטרנט - רצח הנער אופיר רחום (17 ינואר) 2001". Shin Bet. Retrieved 29 September 2012.
  9. ^ "Ofir Rahum". The Israeli Ministry of Foreign Affairs. Retrieved 29 September 2012.
  10. ^ "The Internet lure – the murder of Ofir Rahum, Jan. 17, 2001". Shin Bet. Retrieved 29 September 2012.
  11. ^ נענע10 - נמשכת חקירת רצח הנער אופיר רחום - חדשות
  12. ^ "BBC News - MIDDLE EAST - Woman denies internet murder". Retrieved 16 December 2014.
  13. ^ "ynet החשודה ברצח אופיר רחום מכחישה כל מעורבות ברצח - חדשות". ynet. Retrieved 16 December 2014.
  14. ^ "BBC News - MIDDLE EAST - Woman 'confesses to internet murder'". Retrieved 16 December 2014.
  15. ^ "Senior Fatah-Tanzim operatives responsible for murder of Ofir Rahum-25-Feb-2001". GxMSDev. Retrieved 16 December 2014.
  16. ^ "ynet "הם הצליחו להרוג את אופיר בגלל האינטרנט" - חדשות". ynet. Retrieved 16 December 2014.
  17. ^ "אופיר רחום הובא למנוחות". וואלה! חדשות. Retrieved 16 December 2014.
  18. ^
  19. ^ Letter dated 23 January 2001 from the Permanent Representative of Israel to the United Nations addressed to the Secretary-General
  20. ^ "ynet אמנה מונא הורשעה בגרימת מותו של אופיר רחום - חדשות". ynet. Retrieved 16 December 2014.
  21. ^ "ynet מאסר עולם לאמנה מונא על רצח הנער אופיר רחום - חדשות". ynet. Retrieved 16 December 2014.
  22. ^ Ben-Zur, Raanan (8 June 2006). "Female terrorist attacks fellow inmate". Ynetnews. Retrieved 31 May 2013.
  23. ^ "שרות בתי הסוהר". Retrieved 16 December 2014.
  24. ^ "The reasons for Hamas’s ‘flexibility’ on Schalit swap". Retrieved 16 December 2014.
  25. ^ Levy, Elior (21 December 2011). "Abbas meets freed Palestinian prisoners in Ankara". YNET. Retrieved 21 December 2011.
  26. ^ ABU TOAMEH, KHALED (21 December 2011). "Abbas putting extremist murderers on pedestal". The Jerusalem Post. Retrieved 21 December 2011.

Further reading[edit]

External links[edit]


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From Wikipedia, the free encyclopedia
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Jarre Nabeul.JPG
Nabeul is located in Tunisia
Location in Tunisia
Coordinates: 36°27′15″N 10°44′5″E / 36.45417°N 10.73472°E / 36.45417; 10.73472Coordinates: 36°27′15″N 10°44′5″E / 36.45417°N 10.73472°E / 36.45417; 10.73472
Country Flag of Tunisia.svg Tunisia
Governorate Nabeul Governorate
Population (2014)
 • Total 73,128
Time zone CET (UTC1)

Nabeul (Tunisian Arabic: نابلAbout this sound Nābil ; French: [naˈbøl]) is a coastal town in northeastern Tunisia, on the south coast near to the Cap Bon peninsula. It is located at around 36°27′N 10°44′E / 36.450°N 10.733°E / 36.450; 10.733 and is the capital of the Nabeul Governorate. The city has a population of 73,128 (2014 census).

Nabeul was founded in the 5th century BC by the Greeks of Cyrene, serving as a trade port. Its name is an Arabised form of the Greek Neapolis 'new city' (an etymology it shares with Naples, Neapoli, and Nablus).

It is a popular tourist destination and the main centre of the Tunisian pottery industry.

Twin towns[edit]


text:
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The districts in Nordland county
  Ofoten district
Note: This map does not correspond to the traditional 5 regions of Nordland, since Helgeland here is divided into 3 districts

Ofoten is a traditional district in Nordland county in Northern Norway. It consists of the municipalities of Tysfjord, Ballangen, Evenes, Tjeldsund, Narvik, and Lødingen. It is named after the main fjord, Ofotfjord, which is at the center of this district. The 6,542-square-kilometre (2,526 sq mi) district was home to 30,421 residents in 2005, with almost half of the residents living in the town of Narvik.[1]

View of the Beisfjorden

Ofoten is characterized by fjords surrounded by many mountains with high, jagged peaks reaching up to 1,900 metres (6,200 ft) in height. About 43% of the area lies above 600 metres (2,000 ft). Below elevations of 500 metres (1,600 ft) there are forests. Only 11% of the land is below an elevation of 60 metres (200 ft). The mountains are high especially in the east. In the interior parts of the municipality, there are a number of glaciers, such as Gihtsejiegŋa, and also many lakes. Besides the Ofotfjord, there are a number of other fjords that cut into the landscapes, often with steep shorelines.

The traditional district of Lofoten lies to the west of Ofoten, to the south is the traditional district of Salten, to the east is northern Sweden, and to the north is Troms county.

The municipality of Ofoten was established on 1 January 1838 (see formannskapsdistrikt). On 1 January 1884, the municipality ceased to exist when it was split into two municipalities: Ankenes (population: 1,734) and Evindnæs (population: 2,397).[2]

The district is named after Ofotfjord (Old Norse: Ófóti). The meaning of the first element is unknown and the last element is derived from the Old Norse word fótr which means "foot". The oldest form of the name could have been Úffóti. In this case, the first element is úfr which means "Eurasian eagle-owl". The three inner branches of the Ofotfjord might have been compared with the three claws of an owl.[3]

See also: Lofoten

Media gallery[edit]

  1. ^ Store norske leksikon. "Ofoten. – distrikt" (in Norwegian). Retrieved 2012-05-11.
  2. ^ Jukvam, Dag (1999). "Historisk oversikt over endringer i kommune- og fylkesinndelingen" (PDF) (in Norwegian). Statistisk sentralbyrå.
  3. ^ Store norske leksikon. "Ofoten. – kommune" (in Norwegian). Retrieved 2012-05-11.

Coordinates: 68°18′30″N 16°01′07″E / 68.3083°N 16.0186°E / 68.3083; 16.0186


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Prime number

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"Prime" redirects here. For other uses, see Prime (disambiguation).

A prime number (or a prime) is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example, 5 is prime because 1 and 5 are its only positive integer factors, whereas 6 is composite because it has the divisors 2 and 3 in addition to 1 and 6. The fundamental theorem of arithmetic establishes the central role of primes in number theory: any integer greater than 1 can be expressed as a product of primes that is unique up to ordering. The uniqueness in this theorem requires excluding 1 as a prime because one can include arbitrarily many instances of 1 in any factorization, e.g., 3, 1 · 3, 1 · 1 · 3, etc. are all valid factorizations of 3.

The property of being prime (or not) is called primality. A simple but slow method of verifying the primality of a given number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and \sqrt{n}. Algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of special forms, such as Mersenne numbers. As of April 2014, the largest known prime number has 17,425,170 decimal digits.

There are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no known useful formula that sets apart all of the prime numbers from composites. However, the distribution of primes, that is to say, the statistical behaviour of primes in the large, can be modelled. The first result in that direction is the prime number theorem, proven at the end of the 19th century, which says that the probability that a given, randomly chosen number n is prime is inversely proportional to its number of digits, or to the logarithm of n.

Many questions regarding prime numbers remain open, such as Goldbach's conjecture (that every even integer greater than 2 can be expressed as the sum of two primes), and the twin prime conjecture (that there are infinitely many pairs of primes whose difference is 2). Such questions spurred the development of various branches of number theory, focusing on analytic or algebraic aspects of numbers. Primes are used in several routines in information technology, such as public-key cryptography, which makes use of properties such as the difficulty of factoring large numbers into their prime factors. Prime numbers give rise to various generalizations in other mathematical domains, mainly algebra, such as prime elements and prime ideals.

Definition and examples

A natural number (i.e. 1, 2, 3, 4, 5, 6, etc.) is called a prime number (or a prime) if it has exactly two positive divisors, 1 and the number itself.[1] Natural numbers greater than 1 that are not prime are called composite.

The number 12 is not a prime, as 12 items can be placed into 3 equal-size columns of 4 each (among other ways). 11 items cannot be all placed into several equal-size columns of more than 1 item each without some extra items leftover (a remainder). Therefore, the number 11 is a prime.

Among the numbers 1 to 6, the numbers 2, 3, and 5 are the prime numbers, while 1, 4, and 6 are not prime. 1 is excluded as a prime number, for reasons explained below. 2 is a prime number, since the only natural numbers dividing it are 1 and 2. Next, 3 is prime, too: 1 and 3 do divide 3 without remainder, but 3 divided by 2 gives remainder 1. Thus, 3 is prime. However, 4 is composite, since 2 is another number (in addition to 1 and 4) dividing 4 without remainder:

4 = 2 · 2.

5 is again prime: none of the numbers 2, 3, or 4 divide 5. Next, 6 is divisible by 2 or 3, since

6 = 2 · 3.

Hence, 6 is not prime. The image at the right illustrates that 12 is not prime: 12 = 3 · 4. No even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1, 2, and n. This implies that n is not prime. Accordingly, the term odd prime refers to any prime number greater than 2. Similarly, when written in the usual decimal system, all prime numbers larger than 5 end in 1, 3, 7, or 9, since even numbers are multiples of 2 and numbers ending in 0 or 5 are multiples of 5.

If n is a natural number, then 1 and n divide n without remainder. Therefore, the condition of being a prime can also be restated as: a number is prime if it is greater than one and if none of

2, 3, ..., n − 1

divides n (without remainder). Yet another way to say the same is: a number n > 1 is prime if it cannot be written as a product of two integers a and b, both of which are larger than 1:

n = a · b.

In other words, n is prime if n items cannot be divided up into smaller equal-size groups of more than one item.

The set of all primes is often denoted by P.

The first 168 prime numbers (all the prime numbers less than 1000) are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997 (sequence A000040 in OEIS).

Fundamental theorem of arithmetic

The crucial importance of prime numbers to number theory and mathematics in general stems from the fundamental theorem of arithmetic, which states that every integer larger than 1 can be written as a product of one or more primes in a way that is unique except for the order of the prime factors.[2] Primes can thus be considered the “basic building blocks” of the natural numbers. For example:

23244 = 2 · 2 · 3 · 13 · 149
= 22 · 3 · 13 · 149. (22 denotes the square or second power of 2.)

As in this example, the same prime factor may occur multiple times. A decomposition:

n = p1 · p2 · ... · pt

of a number n into (finitely many) prime factors p1, p2, ... to pt is called prime factorization of n. The fundamental theorem of arithmetic can be rephrased so as to say that any factorization into primes will be identical except for the order of the factors. So, albeit there are many prime factorization algorithms to do this in practice for larger numbers, they all have to yield the same result.

If p is a prime number and p divides a product ab of integers, then p divides a or p divides b. This proposition is known as Euclid's lemma.[3] It is used in some proofs of the uniqueness of prime factorizations.

Primality of one

Most early Greeks did not even consider 1 to be a number,[4] so they could not consider it to be a prime. By the Middle Ages and Renaissance many mathematicians included 1 as the first prime number.[5] In the mid-18th century Christian Goldbach listed 1 as the first prime in his famous correspondence with Leonhard Euler -- who did not agree.[6] In the 19th century many mathematicians still considered the number 1 to be a prime. For example, Derrick Norman Lehmer's list of primes up to 10,006,721, reprinted as late as 1956,[7] started with 1 as its first prime.[8] Henri Lebesgue is said to be the last professional mathematician to call 1 prime.[9] By the early 20th century, mathematicians began to accept that 1 is not a prime number, but rather forms its own special category as a "unit."[10]

A large body of mathematical work would still be valid when calling 1 a prime, but Euclid's fundamental theorem of arithmetic (mentioned above) would not hold as stated. For example, the number 15 can be factored as 3 · 5 and 1 · 3 · 5; if 1 were admitted as a prime, these two presentations would be considered different factorizations of 15 into prime numbers, so the statement of that theorem would have to be modified. Similarly, the sieve of Eratosthenes would not work correctly if 1 were considered a prime: a modified version of the sieve that considers 1 as prime would eliminate all multiples of 1 (that is, all other numbers) and produce as output only the single number 1. Furthermore, the prime numbers have several properties that the number 1 lacks, such as the relationship of the number to its corresponding value of Euler's totient function or the sum of divisors function.[11][12]

The Sieve of Eratosthenes is a simple algorithm for finding all prime numbers up to a specified integer. It was created in the 3rd century BC by Eratosthenes, an ancient Greek mathematician.

There are hints in the surviving records of the ancient Egyptians that they had some knowledge of prime numbers: the Egyptian fraction expansions in the Rhind papyrus, for instance, have quite different forms for primes and for composites. However, the earliest surviving records of the explicit study of prime numbers come from the Ancient Greeks. Euclid's Elements (circa 300 BC) contain important theorems about primes, including the infinitude of primes and the fundamental theorem of arithmetic. Euclid also showed how to construct a perfect number from a Mersenne prime. The Sieve of Eratosthenes, attributed to Eratosthenes, is a simple method to compute primes, although the large primes found today with computers are not generated this way.

After the Greeks, little happened with the study of prime numbers until the 17th century. In 1640 Pierre de Fermat stated (without proof) Fermat's little theorem (later proved by Leibniz and Euler). Fermat also conjectured that all numbers of the form 22n + 1 are prime (they are called Fermat numbers) and he verified this up to n = 4 (or 216 + 1). However, the very next Fermat number 232 + 1 is composite (one of its prime factors is 641), as Euler discovered later, and in fact no further Fermat numbers are known to be prime. The French monk Marin Mersenne looked at primes of the form 2p − 1, with p a prime. They are called Mersenne primes in his honor.

Euler's work in number theory included many results about primes. He showed the infinite series 1/2 + 1/3 + 1/5 + 1/7 + 1/11 + … is divergent. In 1747 he showed that the even perfect numbers are precisely the integers of the form 2p−1(2p − 1), where the second factor is a Mersenne prime.

At the start of the 19th century, Legendre and Gauss independently conjectured that as x tends to infinity, the number of primes up to x is asymptotic to x/ln(x), where ln(x) is the natural logarithm of x. Ideas of Riemann in his 1859 paper on the zeta-function sketched a program that would lead to a proof of the prime number theorem. This outline was completed by Hadamard and de la Vallée Poussin, who independently proved the prime number theorem in 1896.

Proving a number is prime is not done (for large numbers) by trial division. Many mathematicians have worked on primality tests for large numbers, often restricted to specific number forms. This includes Pépin's test for Fermat numbers (1877), Proth's theorem (around 1878), the Lucas–Lehmer primality test (originated 1856),[13] and the generalized Lucas primality test. More recent algorithms like APRT-CL, ECPP, and AKS work on arbitrary numbers but remain much slower.

For a long time, prime numbers were thought to have extremely limited application outside of pure mathematics.[14] This changed in the 1970s when the concepts of public-key cryptography were invented, in which prime numbers formed the basis of the first algorithms such as the RSA cryptosystem algorithm.

Since 1951 all the largest known primes have been found by computers. The search for ever larger primes has generated interest outside mathematical circles. The Great Internet Mersenne Prime Search and other distributed computing projects to find large primes have become popular, while mathematicians continue to struggle with the theory of primes.

Number of prime numbers

Main article: Euclid's theorem

There are infinitely many prime numbers. Another way of saying this is that the sequence

2, 3, 5, 7, 11, 13, ...

of prime numbers never ends. This statement is referred to as Euclid's theorem in honor of the ancient Greek mathematician Euclid, since the first known proof for this statement is attributed to him. Many more proofs of the infinitude of primes are known, including an analytical proof by Euler, Goldbach's proof based on Fermat numbers,[15] Furstenberg's proof using general topology,[16] and Kummer's elegant proof.[17]

Euclid's proof

Euclid's proof (Book IX, Proposition 20[18]) considers any finite set S of primes. The key idea is to consider the product of all these numbers plus one:

 N = 1 + \prod_{p\in S} p.

Like any other natural number, N is divisible by at least one prime number (it is possible that N itself is prime).

None of the primes by which N is divisible can be members of the finite set S of primes with which we started, because dividing N by any one of these leaves a remainder of 1. Therefore, the primes by which N is divisible are additional primes beyond the ones we started with. Thus any finite set of primes can be extended to a larger finite set of primes.

It is often erroneously reported that Euclid begins with the assumption that the set initially considered contains all prime numbers, leading to a contradiction, or that it contains precisely the n smallest primes rather than any arbitrary finite set of primes.[19] Today, the product of the smallest n primes plus 1 is conventionally called the nth Euclid number.

Euler's analytical proof

Euler's proof uses the sum of the reciprocals of primes,

S(p) = \frac 1 2 + \frac 1 3 + \frac 1 5 + \frac 1 7 + \cdots + \frac 1 p.

This sum becomes larger than any arbitrary real number provided that p is big enough.[20] This shows that there are infinitely many primes, since otherwise this sum would grow only until the biggest prime p is reached. The growth of S(p) is quantified by Mertens' second theorem.[21] For comparison, the sum

\frac 1 {1^2} + \frac 1 {2^2} + \frac 1 {3^2} + \cdots + \frac 1 {n^2} = \sum_{i=1}^n \frac 1 {i^2}

does not grow to infinity as n goes to infinity (see Basel problem). In this sense, prime numbers occur more often than squares of natural numbers. Brun's theorem states that the sum of the reciprocals of twin primes,

 \left( {\frac{1}{3} + \frac{1}{5}} \right) + \left( {\frac{1}{5} + \frac{1}{7}} \right) + \left( {\frac{1}{{11}} + \frac{1}{{13}}} \right) +  \cdots = \sum\limits_{ \begin{smallmatrix} p \text{ prime, } \\ p + 2 \text { prime} \end{smallmatrix}} {\left( {\frac{1}{p} + \frac{1}{{p + 2}}} \right)},

is finite.

Testing primality and integer factorization

There are various methods to determine whether a given number n is prime. The most basic routine, trial division, is of little practical use because of its slowness. One group of modern primality tests is applicable to arbitrary numbers, while more efficient tests are available for particular numbers. Most such methods only tell whether n is prime or not. Routines also yielding one (or all) prime factors of n are called factorization algorithms.

Trial division

The most basic method of checking the primality of a given integer n is called trial division. This routine consists of dividing n by each integer m that is greater than 1 and less than or equal to the square root of n. If the result of any of these divisions is an integer, then n is not a prime, otherwise it is a prime. Indeed, if n=a b is composite (with a and b ≠ 1) then one of the factors a or b is necessarily at most \sqrt{n}. For example, for  n = 37 , the trial divisions are by m = 2, 3, 4, 5, and 6. None of these numbers divides 37, so 37 is prime. This routine can be implemented more efficiently if a complete list of primes up to \sqrt{n} is known—then trial divisions need to be checked only for those m that are prime. For example, to check the primality of 37, only three divisions are necessary (m = 2, 3, and 5), given that 4 and 6 are composite.

While a simple method, trial division quickly becomes impractical for testing large integers because the number of possible factors grows too rapidly as n increases. According to the prime number theorem explained below, the number of prime numbers less than \sqrt{n} is approximately given by \sqrt{n} / \ln(\sqrt{n}), so the algorithm may need up to this number of trial divisions to check the primality of n. For n = 1020, this number is 450 million—too large for many practical applications.

An algorithm yielding all primes up to a given limit, such as required in the primes-only trial division method, is called a prime number sieve. The oldest example, the sieve of Eratosthenes (see above), is still the most commonly used. The sieve of Atkin is another option. Before the advent of computers, lists of primes up to bounds like 107 were also used.[22]

Primality testing versus primality proving

Modern primality tests for general numbers n can be divided into two main classes, probabilistic (or "Monte Carlo") and deterministic algorithms. Deterministic algorithms provide a way to tell for sure whether a given number is prime or not. For example, trial division is a deterministic algorithm because, if it performed correctly, it will always identify a prime number as prime and a composite number as composite. Probabilistic algorithms are normally faster, but do not completely prove that a number is prime. These tests rely on testing a given number in a partly random way. For example, a given test might pass all the time if applied to a prime number, but pass only with probability p if applied to a composite number. If we repeat the test n times and pass every time, then the probability that our number is composite is 1/(1-p)n, which decreases exponentially with the number of tests, so we can be as sure as we like (though never perfectly sure) that the number is prime. On the other hand, if the test ever fails, then we know that the number is composite.

A particularly simple example of a probabilistic test is the Fermat primality test, which relies on the fact (Fermat's little theorem) that np≡n (mod p) for any n if p is a prime number. If we have a number b that we want to test for primality, then we work out nb (mod b) for a random value of n as our test. A flaw with this test is that there are some composite numbers (the Carmichael numbers) that satisfy the Fermat identity even though they are not prime, so the test has no way of distinguishing between prime numbers and Carmichael numbers. Carmichael numbers are substantially rarer than prime numbers, though, so this test can be useful for practical purposes. More powerful extensions of the Fermat primality test, such as the Baillie-PSW, Miller-Rabin, and Solovay-Strassen tests, are guaranteed to fail at least some of the time when applied to a composite number.

Deterministic algorithms do not erroneously report composite numbers as prime. In practice, the fastest such method is known as elliptic curve primality proving. Analyzing its run time is based on heuristic arguments, as opposed to the rigorously proven complexity of the more recent AKS primality test. Deterministic methods are typically slower than probabilistic ones, so the latter ones are typically applied first before a more time-consuming deterministic routine is employed.

The following table lists a number of prime tests. The running time is given in terms of n, the number to be tested and, for probabilistic algorithms, the number k of tests performed. Moreover, ε is an arbitrarily small positive number, and log is the logarithm to an unspecified base. The big O notation means that, for example, elliptic curve primality proving requires a time that is bounded by a factor (not depending on n, but on ε) times log5+ε(n).

Test Developed in Type Running time Notes
AKS primality test 2002 deterministic O(log6+ε(n))
Elliptic curve primality proving 1977 deterministic O(log5+ε(n)) heuristically
Baillie-PSW primality test 1980 probabilistic O(log3 n) no known counterexamples
Miller–Rabin primality test 1980 probabilistic O(k · log2+ε (n)) error probability 4k
Solovay–Strassen primality test 1977 probabilistic O(k · log3 n) error probability 2k
Fermat primality test probabilistic O(k · log2+ε (n)) fails for Carmichael numbers

Special-purpose algorithms and the largest known prime

Further information: List of prime numbers
Construction of a regular pentagon. 5 is a Fermat prime.

In addition to the aforementioned tests applying to any natural number n, a number of much more efficient primality tests is available for special numbers. For example, to run Lucas' primality test requires the knowledge of the prime factors of n − 1, while the Lucas–Lehmer primality test needs the prime factors of n + 1 as input. For example, these tests can be applied to check whether

n! ± 1 = 1 · 2 · 3 · ... · n ± 1

are prime. Prime numbers of this form are known as factorial primes. Other primes where either p + 1 or p − 1 is of a particular shape include the Sophie Germain primes (primes of the form 2p + 1 with p prime), primorial primes, Fermat primes and Mersenne primes, that is, prime numbers that are of the form 2p − 1, where p is an arbitrary prime. The Lucas–Lehmer test is particularly fast for numbers of this form. This is why the largest known prime has almost always been a Mersenne prime since the dawn of electronic computers.

Fermat primes are of the form

Fk = 22k + 1,

with k an arbitrary natural number. They are named after Pierre de Fermat who conjectured that all such numbers Fk are prime. This was based on the evidence of the first five numbers in this series—3, 5, 17, 257, and 65,537—being prime. However, F5 is composite and so are all other Fermat numbers that have been verified as of 2015. A regular n-gon is constructible using straightedge and compass if and only if

n = 2i · m

where m is a product of any number of distinct Fermat primes and i is any natural number, including zero.

The following table gives the largest known primes of the mentioned types. Some of these primes have been found using distributed computing. In 2009, the Great Internet Mersenne Prime Search project was awarded a US$100,000 prize for first discovering a prime with at least 10 million digits.[23] The Electronic Frontier Foundation also offers $150,000 and $250,000 for primes with at least 100 million digits and 1 billion digits, respectively.[24] Some of the largest primes not known to have any particular form (that is, no simple formula such as that of Mersenne primes) have been found by taking a piece of semi-random binary data, converting it to a number n, multiplying it by 256k for some positive integer k, and searching for possible primes within the interval [256kn + 1, 256k(n + 1) − 1].[citation needed]

Type Prime Number of decimal digits Date Found by
Mersenne prime 257,885,161 − 1 17,425,170 January 25, 2013 Great Internet Mersenne Prime Search
not a Mersenne prime (Proth number) 19,249 × 213,018,586 + 1 3,918,990 March 26, 2007 Seventeen or Bust
factorial prime 150209! + 1 712,355 October 2011 PrimeGrid[25]
primorial prime 1098133# - 1 476,311 March 2012 PrimeGrid[26]
twin primes 3756801695685 × 2666669 ± 1 200,700 December 2011 PrimeGrid[27]

Integer factorization

Main article: Integer factorization

Given a composite integer n, the task of providing one (or all) prime factors is referred to as factorization of n. Elliptic curve factorization is an algorithm relying on arithmetic on an elliptic curve.

In 1975, number theorist Don Zagier commented that primes both

grow like weeds among the natural numbers, seeming to obey no other law than that of chance [but also] exhibit stunning regularity [and] that there are laws governing their behavior, and that they obey these laws with almost military precision.[28]

The distribution of primes in the large, such as the question how many primes are smaller than a given, large threshold, is described by the prime number theorem, but no efficient formula for the n-th prime is known.

There are arbitrarily long sequences of consecutive non-primes, as for every positive integer n the n consecutive integers from (n+1)! + 2 to (n+1)! + n + 1 (inclusive) are all composite (as (n+1)! + k is divisible by k for k between 2 and n + 1).

Dirichlet's theorem on arithmetic progressions, in its basic form, asserts that linear polynomials

p(n) = a + bn\,

with coprime integers a and b take infinitely many prime values. Stronger forms of the theorem state that the sum of the reciprocals of these prime values diverges, and that different such polynomials with the same b have approximately the same proportions of primes.

The corresponding question for quadratic polynomials is less well-understood.

Formulas for primes

Main article: formulas for primes

There is no known efficient formula for primes. For example, Mills' theorem and a theorem of Wright assert that there are real constants A>1 and μ such that

\left \lfloor A^{3^{n}}\right \rfloor \text{ and } \left \lfloor 2^{\dots^{2^{2^\mu}}} \right \rfloor

are prime for any natural number n. Here \lfloor - \rfloor represents the floor function, i.e., largest integer not greater than the number in question. The latter formula can be shown using Bertrand's postulate (proven first by Chebyshev), which states that there always exists at least one prime number p with n < p < 2n − 2, for any natural number n > 3. However, computing A or μ requires the knowledge of infinitely many primes to begin with.[29] Another formula is based on Wilson's theorem and generates the number 2 many times and all other primes exactly once.

There is no non-constant polynomial, even in several variables, that takes only prime values. However, there is a set of Diophantine equations in 9 variables and one parameter with the following property: the parameter is prime if and only if the resulting system of equations has a solution over the natural numbers. This can be used to obtain a single formula with the property that all its positive values are prime.[30]

Number of prime numbers below a given number

A chart depicting π(n) (blue), n / ln (n) (green) and Li(n) (red)

The prime counting function π(n) is defined as the number of primes not greater than n. For example, π(11) = 5, since there are five primes less than or equal to 11. There are known algorithms to compute exact values of π(n) faster than it would be possible to compute each prime up to n. The prime number theorem states that π(n) is approximately given by

\pi(n) \approx \frac n {\ln n},

in the sense that the ratio of π(n) and the right hand fraction approaches 1 when n grows to infinity. This implies that the likelihood that a number less than n is prime is (approximately) inversely proportional to the number of digits in n. A more accurate estimate for π(n) is given by the offset logarithmic integral

\operatorname{Li}(n) = \int_2^n \frac{dt}{\ln t}.

The prime number theorem also implies estimates for the size of the n-th prime number pn (i.e., p1 = 2, p2 = 3, etc.): up to a bounded factor, pn grows like n log(n).[31] In particular, the prime gaps, i.e. the differences pnpn−1 of two consecutive primes, become arbitrarily large. This latter statement can also be seen in a more elementary way by noting that the sequence n! + 2, n! + 3, …, n! + n (for the notation n! read factorial) consists of n − 1 composite numbers, for any natural number n.

Arithmetic progressions

An arithmetic progression is the set of natural numbers that give the same remainder when divided by some fixed number q called modulus. For example,

3, 12, 21, 30, 39, ...,

is an arithmetic progression modulo q = 9. Except for 3, none of these numbers is prime, since 3 + 9n = 3(1 + 3n) so that the remaining numbers in this progression are all composite. (In general terms, all prime numbers above q are of the form q#·n + m, where 0 < m < q#, and m has no prime factor ≤ q.) Thus, the progression

a, a + q, a + 2q, a + 3q, …

can have infinitely many primes only when a and q are coprime, i.e., their greatest common divisor is one. If this necessary condition is satisfied, Dirichlet's theorem on arithmetic progressions asserts that the progression contains infinitely many primes. The picture below illustrates this with q = 9: the numbers are "wrapped around" as soon as a multiple of 9 is passed. Primes are highlighted in red. The rows (=progressions) starting with a = 3, 6, or 9 contain at most one prime number. In all other rows (a = 1, 2, 4, 5, 7, and 8) there are infinitely many prime numbers. What is more, the primes are distributed equally among those rows in the long run—the density of all primes congruent a modulo 9 is 1/6.

Prime numbers (highlighted in red) in arithmetic progression modulo 9.

The Green–Tao theorem shows that there are arbitrarily long arithmetic progressions consisting of primes.[32] An odd prime p is expressible as the sum of two squares, p = x2 + y2, exactly if p is congruent 1 modulo 4 (Fermat's theorem on sums of two squares).

Prime values of quadratic polynomials

The Ulam spiral. Red pixels show prime numbers. Primes of the form 4n2 − 2n + 41 are highlighted in blue.

Euler noted that the function

n^2 + n + 41\,

gives prime numbers for 0 ≤ n < 40,[33][34] a fact leading into deep algebraic number theory, more specifically Heegner numbers. For bigger n, it does take composite values. The Hardy-Littlewood conjecture F makes an asymptotic prediction about the density of primes among the values of quadratic polynomials (with integer coefficients a, b, and c)

f(n) = a x^2 + bx + c\,

in terms of Li(n) and the coefficients a, b, and c. However, progress has proved hard to come by: no quadratic polynomial (with a ≠ 0) is known to take infinitely many prime values. The Ulam spiral depicts all natural numbers in a spiral-like way. Surprisingly, prime numbers cluster on certain diagonals and not others, suggesting that some quadratic polynomials take prime values more often than other ones.

Open questions

Zeta function and the Riemann hypothesis

Main article: Riemann hypothesis
Plot of the zeta function ζ(s). At s=1, the function has a pole, that is to say, it tends to infinity.

The Riemann zeta function ζ(s) is defined as an infinite sum

\zeta(s)=\sum_{n=1}^\infin \frac{1}{n^s},

where s is a complex number with real part bigger than 1. It is a consequence of the fundamental theorem of arithmetic that this sum agrees with the infinite product

\prod_{p \text{ prime}} \frac{1}{1-p^{-s}}.

The zeta function is closely related to prime numbers. For example, the aforementioned fact that there are infinitely many primes can also be seen using the zeta function: if there were only finitely many primes then ζ(1) would have a finite value. However, the harmonic series 1 + 1/2 + 1/3 + 1/4 + ... diverges (i.e., exceeds any given number), so there must be infinitely many primes. Another example of the richness of the zeta function and a glimpse of modern algebraic number theory is the following identity (Basel problem), due to Euler,

\zeta(2) = \prod_{p} \frac{1}{1-p^{-2}}= \frac{\pi^2}{6}.

The reciprocal of ζ(2), 6/π2, is the probability that two numbers selected at random are relatively prime.[35][36]

The unproven Riemann hypothesis, dating from 1859, states that except for s = −2, −4, ..., all zeroes of the ζ-function have real part equal to 1/2. The connection to prime numbers is that it essentially says that the primes are as regularly distributed as possible.[clarification needed] From a physical viewpoint, it roughly states that the irregularity in the distribution of primes only comes from random noise. From a mathematical viewpoint, it roughly states that the asymptotic distribution of primes (about x/log x of numbers less than x are primes, the prime number theorem) also holds for much shorter intervals of length about the square root of x (for intervals near x). This hypothesis is generally believed to be correct. In particular, the simplest assumption is that primes should have no significant irregularities without good reason.

Other conjectures

In addition to the Riemann hypothesis, many more conjectures revolving about primes have been posed. Often having an elementary formulation, many of these conjectures have withstood a proof for decades: all four of Landau's problems from 1912 are still unsolved. One of them is Goldbach's conjecture, which asserts that every even integer n greater than 2 can be written as a sum of two primes. As of February 2011, this conjecture has been verified for all numbers up to n = 2 · 1017.[37] Weaker statements than this have been proven, for example Vinogradov's theorem says that every sufficiently large odd integer can be written as a sum of three primes. Chen's theorem says that every sufficiently large even number can be expressed as the sum of a prime and a semiprime, the product of two primes. Also, any even integer can be written as the sum of six primes.[38] The branch of number theory studying such questions is called additive number theory.

Other conjectures deal with the question whether an infinity of prime numbers subject to certain constraints exists. It is conjectured that there are infinitely many Fibonacci primes[39] and infinitely many Mersenne primes, but not Fermat primes.[40] It is not known whether or not there are an infinite number of Wieferich primes and of prime Euclid numbers.

A third type of conjectures concerns aspects of the distribution of primes. It is conjectured that there are infinitely many twin primes, pairs of primes with difference 2 (twin prime conjecture). Polignac's conjecture is a strengthening of that conjecture, it states that for every positive integer n, there are infinitely many pairs of consecutive primes that differ by 2n.[41] It is conjectured there are infinitely many primes of the form n2 + 1.[42] These conjectures are special cases of the broad Schinzel's hypothesis H. Brocard's conjecture says that there are always at least four primes between the squares of consecutive primes greater than 2. Legendre's conjecture states that there is a prime number between n2 and (n + 1)2 for every positive integer n. It is implied by the stronger Cramér's conjecture.

For a long time, number theory in general, and the study of prime numbers in particular, was seen as the canonical example of pure mathematics, with no applications outside of the self-interest of studying the topic with the exception of use of prime numbered gear teeth to distribute wear evenly. In particular, number theorists such as British mathematician G. H. Hardy prided themselves on doing work that had absolutely no military significance.[43] However, this vision was shattered in the 1970s, when it was publicly announced that prime numbers could be used as the basis for the creation of public key cryptography algorithms. Prime numbers are also used for hash tables and pseudorandom number generators.

Some rotor machines were designed with a different number of pins on each rotor, with the number of pins on any one rotor either prime, or coprime to the number of pins on any other rotor. This helped generate the full cycle of possible rotor positions before repeating any position.

The International Standard Book Numbers work with a check digit, which exploits the fact that 11 is a prime.

Arithmetic modulo a prime and finite fields

Main article: Modular arithmetic

Modular arithmetic modifies usual arithmetic by only using the numbers

\{0, 1, 2, \dots, n-1 \}, \,

where n is a fixed natural number called modulus. Calculating sums, differences and products is done as usual, but whenever a negative number or a number greater than n − 1 occurs, it gets replaced by the remainder after division by n. For instance, for n = 7, the sum 3 + 5 is 1 instead of 8, since 8 divided by 7 has remainder 1. This is referred to by saying "3 + 5 is congruent to 1 modulo 7" and is denoted

3 + 5 \equiv 1 \pmod 7.

Similarly, 6 + 1 ≡ 0 (mod 7), 2 − 5 ≡ 4 (mod 7), since −3 + 7 = 4, and 3 · 4 ≡ 5 (mod 7) as 12 has remainder 5. Standard properties of addition and multiplication familiar from the integers remain valid in modular arithmetic. In the parlance of abstract algebra, the above set of integers, which is also denoted Z/nZ, is therefore a commutative ring for any n. Division, however, is not in general possible in this setting. For example, for n = 6, the equation

3 \cdot x \equiv 2 \pmod 6,

a solution x of which would be an analogue of 2/3, cannot be solved, as one can see by calculating 3 · 0, ..., 3 · 5 modulo 6. The distinctive feature of prime numbers is the following: division is possible in modular arithmetic if and only if n is a prime. Equivalently, n is prime if and only if all integers m satisfying 2 ≤ mn − 1 are coprime to n, i.e. their only common divisor is one. Indeed, for n = 7, the equation

3 \cdot x \equiv 2 \ \ (\operatorname{mod}\ 7),

has a unique solution, x = 3. Because of this, for any prime p, Z/pZ (also denoted Fp) is called a field or, more specifically, a finite field since it contains finitely many, namely p, elements.

A number of theorems can be derived from inspecting Fp in this abstract way. For example, Fermat's little theorem, stating

a^{p-1} \equiv 1 (\operatorname{mod}\ p)

for any integer a not divisible by p, may be proved using these notions. This implies

\sum_{a=1}^{p-1} a^{p-1} \equiv (p-1) \cdot 1 \equiv -1 \pmod p.

Giuga's conjecture says that this equation is also a sufficient condition for p to be prime. Another consequence of Fermat's little theorem is the following: if p is a prime number other than 2 and 5, 1/p is always a recurring decimal, whose period is p − 1 or a divisor of p − 1. The fraction 1/p expressed likewise in base q (rather than base 10) has similar effect, provided that p is not a prime factor of q. Wilson's theorem says that an integer p > 1 is prime if and only if the factorial (p − 1)! + 1 is divisible by p. Moreover, an integer n > 4 is composite if and only if (n − 1)! is divisible by n.

Other mathematical occurrences of primes

Many mathematical domains make great use of prime numbers. An example from the theory of finite groups are the Sylow theorems: if G is a finite group and pn is the highest power of the prime p that divides the order of G, then G has a subgroup of order pn. Also, any group of prime order is cyclic (Lagrange's theorem).

Public-key cryptography

Several public-key cryptography algorithms, such as RSA and the Diffie–Hellman key exchange, are based on large prime numbers (for example 512 bit primes are frequently used for RSA and 1024 bit primes are typical for Diffie–Hellman.). RSA relies on the assumption that it is much easier (i.e., more efficient) to perform the multiplication of two (large) numbers x and y than to calculate x and y (assumed coprime) if only the product xy is known. The Diffie–Hellman key exchange relies on the fact that there are efficient algorithms for modular exponentiation, while the reverse operation the discrete logarithm is thought to be a hard problem.

Prime numbers in nature

The evolutionary strategy used by cicadas of the genus Magicicada make use of prime numbers.[44] These insects spend most of their lives as grubs underground. They only pupate and then emerge from their burrows after 7, 13 or 17 years, at which point they fly about, breed, and then die after a few weeks at most. The logic for this is believed to be that the prime number intervals between emergences make it very difficult for predators to evolve that could specialize as predators on Magicicadas.[45] If Magicicadas appeared at a non-prime number intervals, say every 12 years, then predators appearing every 2, 3, 4, 6, or 12 years would be sure to meet them. Over a 200-year period, average predator populations during hypothetical outbreaks of 14- and 15-year cicadas would be up to 2% higher than during outbreaks of 13- and 17-year cicadas.[46] Though small, this advantage appears to have been enough to drive natural selection in favour of a prime-numbered life-cycle for these insects.

There is speculation[by whom?] that the zeros of the zeta function are connected to the energy levels of complex quantum systems.[47]

The concept of prime number is so important that it has been generalized in different ways in various branches of mathematics. Generally, "prime" indicates minimality or indecomposability, in an appropriate sense. For example, the prime field is the smallest subfield of a field F containing both 0 and 1. It is either Q or the finite field with p elements, whence the name.[48] Often a second, additional meaning is intended by using the word prime, namely that any object can be, essentially uniquely, decomposed into its prime components. For example, in knot theory, a prime knot is a knot that is indecomposable in the sense that it cannot be written as the knot sum of two nontrivial knots. Any knot can be uniquely expressed as a connected sum of prime knots.[49] Prime models and prime 3-manifolds are other examples of this type.

Prime elements in rings

Prime numbers give rise to two more general concepts that apply to elements of any commutative ring R, an algebraic structure where addition, subtraction and multiplication are defined: prime elements and irreducible elements. An element p of R is called prime element if it is neither zero nor a unit (i.e., does not have a multiplicative inverse) and satisfies the following requirement: given x and y in R such that p divides the product xy, then p divides x or y. An element is irreducible if it is not a unit and cannot be written as a product of two ring elements that are not units. In the ring Z of integers, the set of prime elements equals the set of irreducible elements, which is

\{ \dots, -11, -7, -5, -3, -2, 2, 3, 5, 7, 11, \dots \}\, .

In any ring R, any prime element is irreducible. The converse does not hold in general, but does hold for unique factorization domains.

The fundamental theorem of arithmetic continues to hold in unique factorization domains. An example of such a domain is the Gaussian integers Z[i], that is, the set of complex numbers of the form a + bi where i denotes the imaginary unit and a and b are arbitrary integers. Its prime elements are known as Gaussian primes. Not every prime (in Z) is a Gaussian prime: in the bigger ring Z[i], 2 factors into the product of the two Gaussian primes (1 + i) and (1 − i). Rational primes (i.e. prime elements in Z) of the form 4k + 3 are Gaussian primes, whereas rational primes of the form 4k + 1 are not.

Prime ideals

Main article: Prime ideals

In ring theory, the notion of number is generally replaced with that of ideal. Prime ideals, which generalize prime elements in the sense that the principal ideal generated by a prime element is a prime ideal, are an important tool and object of study in commutative algebra, algebraic number theory and algebraic geometry. The prime ideals of the ring of integers are the ideals (0), (2), (3), (5), (7), (11), … The fundamental theorem of arithmetic generalizes to the Lasker–Noether theorem, which expresses every ideal in a Noetherian commutative ring as an intersection of primary ideals, which are the appropriate generalizations of prime powers.[50]

Prime ideals are the points of algebro-geometric objects, via the notion of the spectrum of a ring.[51] Arithmetic geometry also benefits from this notion, and many concepts exist in both geometry and number theory. For example, factorization or ramification of prime ideals when lifted to an extension field, a basic problem of algebraic number theory, bears some resemblance with ramification in geometry. Such ramification questions occur even in number-theoretic questions solely concerned with integers. For example, prime ideals in the ring of integers of quadratic number fields can be used in proving quadratic reciprocity, a statement that concerns the solvability of quadratic equations

x^2 \equiv p \ \ (\text{mod } q),\,

where x is an integer and p and q are (usual) prime numbers.[52] Early attempts to prove Fermat's Last Theorem climaxed when Kummer introduced regular primes, primes satisfying a certain requirement concerning the failure of unique factorization in the ring consisting of expressions

a_0 + a_1 \zeta + \cdots + a_{p-1} \zeta^{p-1}\, ,

where a0, ..., ap−1 are integers and ζ is a complex number such that ζp = 1.[53]

Valuation theory studies certain functions from a field K to the real numbers R called valuations.[54] Every such valuation yields a topology on K, and two valuations are called equivalent if they yield the same topology. A prime of K (sometimes called a place of K) is an equivalence class of valuations. For example, the p-adic valuation of a rational number q is defined to be the integer vp(q), such that

q = p^{v_p(q)} \frac {r}{s},

where both r and s are not divisible by p. For example, v3(18/7) = 2. The p-adic norm is defined as[nb 1]

\left| q \right|_p := p^{-v_p(q)}. \,

In particular, this norm gets smaller when a number is multiplied by p, in sharp contrast to the usual absolute value (also referred to as the infinite prime). While completing Q (roughly, filling the gaps) with respect to the absolute value yields the field of real numbers, completing with respect to the p-adic norm |−|p yields the field of p-adic numbers.[55] These are essentially all possible ways to complete Q, by Ostrowski's theorem. Certain arithmetic questions related to Q or more general global fields may be transferred back and forth to the completed (or local) fields. This local-global principle again underlines the importance of primes to number theory.

In the arts and literature

Prime numbers have influenced many artists and writers. The French composer Olivier Messiaen used prime numbers to create ametrical music through "natural phenomena". In works such as La Nativité du Seigneur (1935) and Quatre études de rythme (1949–50), he simultaneously employs motifs with lengths given by different prime numbers to create unpredictable rhythms: the primes 41, 43, 47 and 53 appear in the third étude, "Neumes rythmiques". According to Messiaen this way of composing was "inspired by the movements of nature, movements of free and unequal durations".[56]

In his science fiction novel Contact, NASA scientist Carl Sagan suggested that prime numbers could be used as a means of communicating with aliens, an idea that he had first developed informally with American astronomer Frank Drake in 1975.[57] In the novel The Curious Incident of the Dog in the Night-Time by Mark Haddon, the narrator arranges the sections of the story by consecutive prime numbers.[58]

Many films, such as Cube, Sneakers, The Mirror Has Two Faces and A Beautiful Mind reflect a popular fascination with the mysteries of prime numbers and cryptography.[59] Prime numbers are used as a metaphor for loneliness and isolation in the Paolo Giordano novel The Solitude of Prime Numbers, in which they are portrayed as "outsiders" among integers.[60]

See also

  1. ^ Some sources also put \left| q \right|_p := e^{-v_p(q)}. \,.
  1. ^ Dudley, Underwood (1978), Elementary number theory (2nd ed.), W. H. Freeman and Co., ISBN 978-0-7167-0076-0 , p. 10, section 2
  2. ^ Dudley 1978, Section 2, Theorem 2
  3. ^ Dudley 1978, Section 2, Lemma 5
  4. ^ See, for example, David E. Joyce's commentary on Euclid's Elements, Book VII, definitions 1 and 2.
  5. ^
  6. ^ Weisstein, Eric W., "Goldbach Conjecture," MathWorld
  7. ^ Riesel 1994, p. 36
  8. ^ Conway & Guy 1996, pp. 129–130
  9. ^ Derbyshire, John (2003), "The Prime Number Theorem", Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics, Washington, D.C.: Joseph Henry Press, p. 33, ISBN 978-0-309-08549-6, OCLC 249210614
  10. ^
  11. ^ ""Arguments for and against the primality of 1".
  12. ^ "Why is the number one not prime?"
  13. ^ The Largest Known Prime by Year: A Brief History Prime Curios!: 17014…05727 (39-digits)
  14. ^ For instance, Beiler writes that number theorist Ernst Kummer loved his ideal numbers, closely related to the primes, "because they had not soiled themselves with any practical applications", and Katz writes that Edmund Landau, known for his work on the distribution of primes, "loathed practical applications of mathematics", and for this reason avoided subjects such as geometry that had already shown themselves to be useful. Beiler, Albert H. (1966), Recreations in the Theory of Numbers: The Queen of Mathematics Entertains, Dover, p. 2, ISBN 9780486210964 . Katz, Shaul (2004), "Berlin roots—Zionist incarnation: the ethos of pure mathematics and the beginnings of the Einstein Institute of Mathematics at the Hebrew University of Jerusalem", Science in Context 17 (1-2): 199–234, doi:10.1017/S0269889704000092, MR 2089305 .
  15. ^ Letter in Latin from Goldbach to Euler, July 1730.
  16. ^ Furstenberg 1955
  17. ^ Ribenboim 2004, p. 4
  18. ^ James Williamson (translator and commentator), The Elements of Euclid, With Dissertations, Clarendon Press, Oxford, 1782, page 63, English translation of Euclid's proof
  19. ^ Hardy, Michael; Woodgold, Catherine (2009). "Prime Simplicity". Mathematical Intelligencer 31 (4): 44–52. doi:10.1007/s00283-009-9064-8.
  20. ^ Apostol, Tom M. (1976), Introduction to Analytic Number Theory, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90163-3 , Section 1.6, Theorem 1.13
  21. ^ Apostol 1976, Section 4.8, Theorem 4.12
  22. ^ (Lehmer 1909).
  23. ^ "Record 12-Million-Digit Prime Number Nets $100,000 Prize". Electronic Frontier Foundation. October 14, 2009. Retrieved 2010-01-04.
  24. ^ "EFF Cooperative Computing Awards". Electronic Frontier Foundation. Retrieved 2010-01-04.
  25. ^ Chris K. Caldwell. "The Top Twenty: Factorial". Retrieved 2013-02-05.
  26. ^ Chris K. Caldwell. "The Top Twenty: Primorial". Retrieved 2013-02-05.
  27. ^ Chris K. Caldwell. "The Top Twenty: Twin Primes". Retrieved 2013-02-05.
  28. ^ Havil 2003, p. 171
  29. ^ p. 15
  30. ^ Matiyasevich, Yuri V. (1999), "Formulas for Prime Numbers", in Tabachnikov, Serge, Kvant Selecta: Algebra and Analysis, II, American Mathematical Society, pp. 13–24, ISBN 978-0-8218-1915-9 .
  31. ^ (Tom M. Apostol 1976), Section 4.6, Theorem 4.7
  32. ^ (Ben Green & Terence Tao 2008).
  33. ^ Hua (2009), pp. 176–177"
  34. ^ See list of values, calculated by Wolfram Alpha
  35. ^ Caldwell, Chris. "What is the probability that gcd(n,m)=1?". The Prime Pages. Retrieved 2013-09-06.
  36. ^ C. S. Ogilvy & J. T. Anderson Excursions in Number Theory, pp. 29–35, Dover Publications Inc., 1988 ISBN 0-486-25778-9
  37. ^ Tomás Oliveira e Silva (2011-04-09). "Goldbach conjecture verification". Retrieved 2011-05-21.
  38. ^ Ramaré, O. (1995), "On šnirel'man's constant", Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie IV 22 (4): 645–706, retrieved 2008-08-22.
  39. ^ Caldwell, Chris, The Top Twenty: Lucas Number at The Prime Pages.
  40. ^ E.g., see Guy 1981, problem A3, pp. 7–8
  41. ^ Tattersall, J.J. (2005), Elementary number theory in nine chapters, Cambridge University Press, ISBN 978-0-521-85014-8 , p. 112
  42. ^ Weisstein, Eric W., "Landau's Problems", MathWorld.
  43. ^ Hardy 1940 "No one has yet discovered any warlike purpose to be served by the theory of numbers or relativity, and it seems unlikely that anyone will do so for many years."
  44. ^ Goles, E.; Schulz, O.; Markus, M. (2001). "Prime number selection of cycles in a predator-prey model". Complexity 6 (4): 33–38. doi:10.1002/cplx.1040.
  45. ^ Paulo R. A. Campos, Viviane M. de Oliveira, Ronaldo Giro, and Douglas S. Galvão. (2004), "Emergence of Prime Numbers as the Result of Evolutionary Strategy", Physical Review Letters 93 (9): 098107, arXiv:q-bio/0406017, Bibcode:2004PhRvL..93i8107C, doi:10.1103/PhysRevLett.93.098107.
  46. ^ "Invasion of the Brood". The Economist. May 6, 2004. Retrieved 2006-11-26.
  47. ^ Ivars Peterson (June 28, 1999). "The Return of Zeta". MAA Online. Archived from the original on October 20, 2007. Retrieved 2008-03-14.
  48. ^ Lang, Serge (2002), Algebra, Graduate Texts in Mathematics 211, Berlin, New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556 , Section II.1, p. 90
  49. ^ Schubert, H. "Die eindeutige Zerlegbarkeit eines Knotens in Primknoten". S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl. 1949 (1949), 57–104.
  50. ^ Eisenbud 1995, section 3.3.
  51. ^ Shafarevich, Basic Algebraic Geometry volume 2 (Schemes and Complex Manifolds), p. 5, section V.1
  52. ^ Neukirch, Algebraic Number theory, p. 50, Section I.8
  53. ^ Neukirch, Algebraic Number theory, p. 38, Section I.7
  54. ^ Endler, Valuation Theory, p. 1
  55. ^ Gouvea: p-adic numbers: an introduction, Chapter 3, p. 43
  56. ^ Hill, ed. 1995
  57. ^ Carl Pomerance, Prime Numbers and the Search for Extraterrestrial Intelligence, Retrieved on December 22, 2007
  58. ^ Mark Sarvas, Book Review: The Curious Incident of the Dog in the Night-Time, at The Modern Word, Retrieved on March 30, 2012
  59. ^ The music of primes, Marcus du Sautoy's selection of films featuring prime numbers.
  60. ^ "Introducing Paolo Giordano". Books Quarterly. [dead link]

External links

Prime number generators and calculators


text:
Prince Shōtoku

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Prince Shōtoku
Prince Shotoku.jpg
Prince Shotoku flanked by younger brother (left: Prince Eguri) and first son (right: Prince Yamashiro), drawn by unknown author.[1]
Father Emperor Yōmei
Mother Princess Anahobe no Hashihito
Born (572-02-07)7 February 572
Died 8 April 622(622-04-08) (aged 50)

Prince Shōtoku (聖徳太子 Shōtoku Taishi?, February 7, 572 – April 8, 622[2]), also known as Prince Umayado (厩戸皇子 Umayado no ōji?) or Prince Kamitsumiya (上宮皇子 Kamitsumiya no ōji?), was a semi-legendary regent and a politician of the Asuka period in Japan who served under Empress Suiko. He was the son of Emperor Yōmei and his consort, Princess Anahobe no Hashihito, who was also Yōmei's younger half-sister. His parents were relatives of the ruling Soga clan[3] and he was involved in the defeat of the rival Mononobe clan.[4] The primary source of the life and accomplishments of Prince Shōtoku comes from the Nihon Shoki.

Over successive generations, a devotional cult arose around the figure of Prince Shōtoku for the protection of Japan, the Imperial Family, and for Buddhism. Key religious figures such as Saichō, Shinran and others claimed inspiration or visions attributed to Prince Shōtoku.[4]

Cultural and political role[edit]

Shotoku Taishi by Kogan Zenji

Shōtoku was appointed as regent (Sesshō) in 593 by Empress Suiko, his aunt.[citation needed] Shōtoku, inspired by Buddha's teachings, succeeded in establishing a centralized government during his reign. In 603, he established the Twelve Level Cap and Rank System at the court. He is credited with promulgating a Seventeen-article constitution.

The Prince was an ardent Buddhist and is traditionally attributed the authorship of the Sangyō Gisho or "Annotated Commentaries on the Three Sutras" (the Lotus Sutra, the Vimalakirti Sutra, and the Śrīmālādevī Siṃhanāda Sūtra). The first of these commentaries, Hokke Gisho, is traditionally dated to 615 and thus regarded as "the first Japanese text", in turn making Shōtoku the first Japanese writer.

A legend claims that when Bodhidharma came to Japan, he met with Prince Shōtoku whilst under the guise of a starving beggar. The Prince asked the beggar to identify himself, but the man did not reply. Instead of going ahead, Shōtoku gave him food, drink, and covered him with his purple garment, telling him to "lie in peace". The Prince then sang for the starving man.

Alas! For The wayfarer lying And hungered for rice On the hill of Kataoka (The sunshiny) Art thou become Parentless? Hast thou no lord Flourishing as a bamboo? Alas! For The wayfarer lying And hungered for rice!

The second day, the Prince sent a messenger to the starving man, but he was already dead. Hereupon, Shōtoku was greatly grieved and ordered his burial. Shōtoku later thought the man was no ordinary man for sure, and sending another messenger, discovered the earth had not been disturbed. On opening the tomb there was no body inside, and the Prince's purple garment lay folded on the coffin. The Prince then sent another messenger to claim the garment, and he continued to wear it just as before. Struck by awe, the people praised the Prince "How true it is that a sage knoweth a sage." This legend is linked with the temple of Daruma-dera in Ōji, Nara, where a stone stupa was found underground, which is exceedingly rare.

Prince Shōtoku commissioned the Shitennō-ji (temple) in Settsu Province (present-day Osaka) after his military victory against the powerful Mononobe clan, for he is said to have summoned them to crush his ennemies. Shōtoku's name has been linked with Hōryū-ji, a temple in Yamato Province, and numerous other temples in the Kansai region. Documentation at Hōryū-ji claims that Suiko and Shōtoku founded the temple in the year 607. Archaeological excavations in 1939 have confirmed that Prince Shōtoku's palace, the Ikaruga no miya (斑鳩宮), stood in the eastern part of the current temple complex, where the Tō-in (東院) sits today.[5]

Despite being credited as the founder of Japanese Buddhism, it is also said that the Prince respected Shinto and never visited Buddhist temples without visiting Shinto shrines.[6]

In his correspondence with Emperor Yang of Sui, the Prince's letter contains the earliest written instance in which the Japanese archipelago is named Nihon. The Sui Emperor dispatched a message in 605 that said, "the sovereign of Sui respectfully inquires about the sovereign of Wa."[7] Shōtoku responded by sponsoring a mission led by Ono no Imoko in 607: "From the sovereign of the land of the rising sun (nihon/hi izuru) to the sovereign of the land of the setting sun."[8]

He is said to have been buried at Shinaga in Kawachi Province (modern Osaka Prefecture).[9]

A number of institutes are named after him, such as Shotoku Gakuen University and its associated junior college (both in Gifu). The first syllable of his name (聖), can be read shō in Go'on and can also be read sei in Kan’on. The later reading is found in Seitoku University and its associated junior college (both in Matsudo, Chiba) as well as Tokyo's defunct Seitoku Junior College of Nutrition (and indirectly its replacement Seiei College).

Titles and name[edit]

Shōtoku is known by several titles, although his real name is Prince Umayado (厩戸皇子 Umayado no ōji?, literally ‘the prince of the stable door’) since he was born in front of a stable.[10] He is also known as Toyotomimi (豊聡耳?) or Kamitsumiyaō (上宮王?). In the Kojiki, his name appears as Kamitsumiya no Umayado no Toyotomimi no Mikoto (上宮之厩戸豊聡耳命?). In the Nihon Shoki, in addition to Umayado no ōji, he is referred to as Toyomimito Shōtoku (豊耳聡聖徳), Toyotomimi no Nori no Ōkami (豊聡耳法大王), and simply Nori no Ushi no Ōkami (法主王).

The name by which he is best known today, Prince Shōtoku, first appeared in Kaifūsō, written more than 100 years after his death in 751.

See also[edit]

  1. ^ Binyon, Laurence (2006). Painting in the Far East: An Introduction to the History of Pictorial Art in Asia, Especially China and Japan. p. 85. ISBN 0-543-94830-7. The author of this portrait is unknown; it is generally held to be the work of Korean artist, but is quite probably the work of native hand.
  2. ^ A History of Japan, R.H.P. Mason and J.G. Caiger, Charles E.Tuttle Company, Inc., Tokyo 1977, 0221-000349-4615
  3. ^
  4. ^ a b Como, Michael I. (2006). Shōtoku: ethnicity, ritual, and violence in the Japanese Buddhist tradition. New York: Oxford University Press. ISBN 0-19-518861-6.
  5. ^ John Whitney Hall (1988). "The Asuka Enlightenment". The Cambridge History of Japan. Cambridge University. p. 175. Retrieved 2007-04-03.
  6. ^ Shoichi Watanabe (Professor Emeritus at Sophia University) (2014), 教育提言：私が伝えたい天皇・皇室のこと[My opinion concerning education: What I must hand down regarding the Emperor and the Imperial Family of Japan]. In Seiron, 508, 204-211.
  7. ^ Varley, Paul. (1980). Jinnō Shōtōki, p. 128.
  8. ^ Varley, Paul. (1973). Japanese Culture: A Short History. p. 15
  9. ^ Guth, Christine. "The Divine Boy in Japanese Art." Monumenta Nipponica 42:1 (1987). p12.
  10. ^

  • Como, Michael A. Shotoku: Ethnicity, Ritual and Violence in the Japanese Buddhist Tradition. New York: Oxford University Press, 2008. ISBN 978-0-19-518861-5
  • Varley, H. Paul. (1973). Japanese Culture: A Short History. New York: Praeger Publishers.
  • Varley, Paul. (1980). Jinnō Shōtōki: A Chronicle of Gods and Sovereigns. New York: Columbia University Press. ISBN 978-0-231-04940-5; OCLC 59145842

External links[edit]

Media related to Prince Shōtoku at Wikimedia Commons


text:
From Wikipedia, the free encyclopedia
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Proustite - Chanarcillo, Copiapo Province, Atacama Region, Chile.jpg
Proustite on matrix, crystal size: 1 cm, Chañarcillo district, Chile
Category Sulfosalt mineral
(repeating unit)
Strunz classification 02.GA.05 Neso-sulfarsenites
Dana classification Proustite group
Crystal symmetry H-M Symbol (32/m)
Unit cell a = 10.79 Å, c = 8.69 Å Z = 6
Color Scarlet-vermilion
Crystal habit Crystals prismatic and scalenohedral, massive, compact
Crystal system Trigonal - Hexagonal Scalenohedral Space Group: R 3c
Twinning Common
Cleavage Distinct on {1011}
Fracture Conchoidal to uneven
Tenacity Brittle
Mohs scale hardness 2 – 2.5
Luster Adamantine
Streak Vermilion
Diaphaneity Translucent, darkens when exposed to light
Specific gravity 5.57 measured, 5.625 calculated
Optical properties Uniaxial (-)
Refractive index nω = 3.087 - 3.088 nε = 2.792
Birefringence δ = 0.295 - 0.296
Pleochroism Moderate; cochineal-red to blood-red
References [1][2][3]

Proustite is a sulfosalt mineral consisting of; silver sulfarsenide, Ag3AsS3, known also as light red silver or ruby silver ore, and an important source of the metal. It is closely allied to the corresponding sulfantimonide, pyrargyrite, from which it was distinguished by the chemical analyses of Joseph L. Proust (1754-1826) in 1804, after whom the mineral received its name.

The prismatic crystals are often terminated by the scalenohedron and the obtuse rhombohedron, thus resembling calcite (dog-tooth-spar) in habit. The color is scarlet-vermilion and the lustre adamantine; crystals are transparent and very brilliant, but on exposure to light they soon become dull black and opaque. The streak is scarlet, the hardness 2.5, and the specific gravity 5.57.

Proustite occurs in hydrothermal deposits as a phase in the oxidized and supergene zone. It is associated with other silver minerals and sulfides such as native silver, native arsenic, xanthoconite, stephanite, acanthite, tetrahedrite and chlorargyrite.[1]

Magnificent groups of large crystals have been found at Chañarcillo in Chile; other localities which have yielded fine specimens are Freiberg and Marienberg in Saxony, Joachimsthal in Bohemia and Markirch in Alsace.

See also[edit]

Public Domain This article incorporates text from a publication now in the public domainChisholm, Hugh, ed. (1911). Encyclopædia Britannica (11th ed.). Cambridge University Press.