PolyAgent/wiki_arabic_en_parallel_20231101 huggingface dataset:
Seth Morgan (politician)

Seth Morgan (politician)
Seth Morgan (born February 26, 1978) is a former Republican member of the Ohio House of Representatives, who represented the 36th District from 2009 to 2010.
Ohio House of Representatives
A former member of the Huber Heights City Council, Morgan made a run for the Ohio House of Representatives in 2008, following term limits for incumbent Arlene Setzer.
He was sworn into his first term on January 5, 2009, in a new Democratic controlled House. A year later, Morgan declared himself as a candidate for Ohio Auditor of State and was favored by many members of the Tea Party movement in Ohio. He was defeated by Delaware County Prosecutor David Yost. He was succeeded by Michael Henne.
Morgan was mentioned as a potential successor to Secretary of State Jon Husted, once Husted vacated his seat in the Ohio Senate. However, the seat went to Peggy Lehner. Morgan went to the Ohio Supreme Court to get school-funding records from the administration of Democratic Gov. Ted Strickland, who turned over thousands of pages of documents. Nothing of substance was done with these documents.
On May 16, 2011, Ohio Governor John Kasich named Morgan as a member of the Ohio Retirement Study Council.
References
External links
Seth Morgan for Auditor of State official campaign website
1978 births
Living people
Republican Party members of the Ohio House of Representatives
21st-century American politicians
American Township, Ohio

American Township, Ohio
American Township is one of the twelve townships of Allen County, Ohio, United States. The 2020 census found 14,538 people in the township.
Geography
Located in the center of the county, it borders the following townships:
Sugar Creek Township - north
Bath Township - east
Perry Township - southeastern corner
Shawnee Township - south
Amanda Township - southwest
Marion Township - northwest
Two municipalities are located in American Township: the village of Elida in the northwest, and part of the city of Lima, the county seat of Allen County, in the southeast.
Name and history
It is the only American Township statewide. American Township was named German Township until 1918 when the township's citizens successfully petitioned for the name change, due to the sinking of the Lusitania. The petition cited Anti-German sentiment as the primary reason for the request.
Demographics
Government
The township is governed by a three-member board of trustees, who are elected in November of odd-numbered years to a four-year term beginning on the following January 1. Two are elected in the year after the presidential election and one is elected in the year before it. There is also an elected township fiscal officer, who serves a four-year term beginning on April 1 of the year after the election, which is held in November of the year before the presidential election. Vacancies in the fiscal officership or on the board of trustees are filled by the remaining trustees.
References
External links
American Township official website
Allen County website
Townships in Allen County, Ohio
1918 establishments in Ohio
Townships in Ohio
1922–23 Nemzeti Bajnokság I

1922–23 Nemzeti Bajnokság I
Statistics of Nemzeti Bajnokság I for the 1922–23 season.
Overview
It was contested by 12 teams, and MTK Hungária FC won the championship.
League standings
Results
References
Hungary - List of final tables (RSSSF)
Nemzeti Bajnokság I seasons
Hun
1922–23 in Hungarian football
هوكي الحقل في الألعاب الأولمبية الصيفية 2020 - رجال التصفيات

هوكي الحقل في الألعاب الأولمبية الصيفية 2020 - رجال التصفيات
تأهل اثنا عشر فريقا للرجال لمنافسات هوكي الحقل في الألعاب الأولمبية الصيفية 2020 (تأجلت إلى 2021 بسبب جائحة COVID-19).
جدول
– اليابان تأهلت بصفتها البلد المضيف وكبطل لبطولة الاتحاد الآسيوي، لذلك تمت إضافة هذه الحصة إلى تصفيات FIH الأولمبية بدلاً من الذهاب إلى وصيف البطولة.
أحداث التصفيات الأولمبية
في الأصل، كان من المقرر أن يشارك اثنا عشر فريقًا في التصفيات الأولمبية. كان من المقرر أن تقسم هذه الفرق إلى ستة أزواج؛ يلعب كل زوج سلسلة من النتائج الإجمالية التي تتكون من مبارتين. الفائز في كل سلسلة يتأهل للأولمبياد. مع فوز اليابان بدورة الألعاب الآسيوية 2018 (وبالتالي التأهل مرتين، مرة كمضيفة ومرة واحدة كأبطال آسيا)، سيكون هناك بدلاً من ذلك 14 فريقًا، 7 منهم سيتأهلون.
مؤهل
The participating teams were confirmed on 29 August 2019 by the الاتحاد الدولي للهوكي.
المباريات
تم إجراء قرعة المواجهات السبع الأخيرة في 9 سبتمبر 2019. أقيمت المواجهات على مباراتين، كلاهما في ملعب الفريق المضيف.
ملاحظات
تصفيات هوكي الحقل مؤهلة إلى الألعاب الأولمبية الصيفية 2020
هوكي الحقل في الألعاب الأولمبية الصيفية 2020 – تصفيات الرجال
هوكي الحقل في الألعاب الأولمبية الصيفية 2020 – مسابقة الرجال
هوكي الحقل في الألعاب الأولمبية الصيفية – تصفيات الرجال
Shahrak-e Darya Sar

Shahrak-e Darya Sar (also Romanized as Shahrak-e Daryā Sar; also known as Daryā Sar) is a village in Barik Rud Rural District, in the Central District of Fereydunkenar County, Mazandaran Province, Iran. In a 2006 census, its population was 61, in 20 families.
References
Populated places in Fereydunkenar County
Lulu James

Lulu James is a British electronic and soul singer.
Life and career
James was born in Tanzania, descended from a Maasai tribe. She moved to the United Kingdom when she was six years old and settled in South Shields in North East England. She enrolled on a music development course, where she met her primary producer and co-writer DomZilla.
James self-released her debut EP, Rope Mirage, in 2012 and subsequently signed to Black Butter Records, which issued her debut single, "Be Safe" / "Stuck". James toured the United Kingdom in late 2012 with labelmates Hostage, Kidnap Kid, and Rudimental. By early 2013, she had signed to RCA Records and released the single "Closer". The single "Sweetest Thing" followed in late 2013, and she toured with Ellie Goulding and Annie Mac Presents. In 2014, she released the single "Beautiful People", with an EP to follow later in the year. In 2015, James served as a vocalist on house duo Gorgon City's UK tour.
James's musical inspirations include India.Arie, Beyoncé, James Blake, Mariah Carey, Diplo, Aretha Franklin, Whitney Houston, Michael Jackson, Gil Scott-Heron, Amy Winehouse, and Jamie xx.
Discography
Singles
Others
2012: Rope Mirage (EP)
2012: "Be Safe"/"Stuck"
2013: "Closer"
2013: "Step by Step"
2014: "Beautiful People"
2016: Colours (EP)
2019: "Lulu James Presents 3rdCultureKid"/"ZIM ZIMMER"
Featured in
2013: "Why Didn't You Call?" (Gang Colours feat. Lulu James)
2014: "We Disappear" (Jon Hopkins feat. Lulu James)
2015: "Loving You" (Lane 8 feat. Lulu James)
2017: "Tear My Heart" (Moon Boots feat LuLu James [Marquis Hawkes Extended Mix])
References
External links
Official website
21st-century Black British women singers
Black Butter Records artists
British electronic musicians
British soul singers
Living people
Maasai people
Musicians from South Shields
RCA Records artists
Tanzanian emigrants to the United Kingdom
1990s births
People from Kilimanjaro Region
تورستن بالم

تورستن بالم هو سائق سيارة سباق ومهندس سويدي، ولد في 23 يوليو 1947 في كريستينه هامن في السويد.
مراجع
أشخاص على قيد الحياة
أشخاص من كريستينه هامن
سائقو سيارات سباق سويديون
سائقو فورمولا 1 سويديون
مواليد 1947
محمد بن محمود الطربزوني

محمد بن محمود الطربزوني فقيه حنفي وأديب , اسمه محمد بن محمود بن صالح بن حسن الطربزوني الحنقي الشهير بالمدني , ولد في المدينة المنورة , وقد كان مدرسا وقيما على الكتب بجامع السليمانية في إسطنبول , له عدد من كتب منها تحفة الاخوان في الحلال والحرام من الحيوان والاتحافات السنية في الأحاديث القدسية ورسالة في بيان ما في الصحاح من الاوهام ورسالة في المثلثات ورسالة في الاضداد وجالب الفرج وسالب الحرج , توفي عام 1200 هـ الموافق 1795
المراجع
علماء دين سنة
وفيات 1200 هـ
وفيات 1795
مطار كالينينغراد

مطار كالينينغراد هو مطار يخدم مدينة كالينينغراد في روسيا، ويبعد عن المدينة حوالي 24 كلم. أغلب رحلاته الجوية هي رحلات داخلية، وجزء من هذا المطار لا يزال منطقة عسكرية خاصة بالقوات المسلحة للاتحاد الروسي. عرف المطار تحديثاً كبيراً بين سنتي 2016 و2018 وذلك استعداداً لتنظيم روسيا لكأس العالم 2018.
شركات الطيران والوجهات
المراجع
قواعد سلاح الجو الروسي
كالينينغراد
مطارات في مقاطعة كالينينغرادسكايا
منشآت في القوات البحرية الروسية
Arvid Strömberg

Arvid Strömberg (born 30 June 1991 in Stockholm, Sweden) is a professional Swedish ice hockey player. He is currently a forward for Djurgårdens IF in Elitserien.
Career statistics
References
External links
1991 births
Djurgårdens IF Hockey players
Swedish ice hockey forwards
Living people
Ice hockey people from Stockholm
Gilles Grondin

Gilles Grondin (3 February 1943 – 18 July 2005) was an educator and a politician from Quebec, Canada. He was a Member of the House of Commons of Canada from 1986 to 1988.
Background
He was born in Shawinigan-Sud, Quebec. He attended Saint-Georges school and Séminaire Sainte-Marie, where he eventually worked as a librarian, and served as a dean of men at the local CEGEP. He received his Bachelor of Library Science (BScLib) from the Université de Montreal.
Local politics
Grondin served as Mayor of Shawinigan-Sud from 1977 to 1985. He did not run for re-election in 1985 and was succeeded by Claude Pinard.
Member of Parliament
On 29 September 1986, Grondin won a by-election and filled the seat left by Jean Chrétien, becoming the Liberal Member for the district of Saint-Maurice. Chrétien had left federal politics in February that year and would stay out of office until he became Liberal party leader and subsequently Prime Minister.
After serving in the remaining months of the 33rd Canadian Parliament, Grondin left federal politics and did not campaign in the 1988 federal election. He was succeeded by Denis Pronovost of the Progressive Conservative Party.
Death
He died in 2005 after several years of dealing with cancer.
References
External links
Gilles Grondin biography at the city of Shawinigan
1943 births
2005 deaths
Members of the House of Commons of Canada from Quebec
Mayors of places in Quebec
Liberal Party of Canada MPs
Séminaire Sainte-Marie alumni
Université de Montréal alumni
Deaths from cancer in Quebec
محمد تقي المامقاني

محمد تقي المامقاني
الميرزا محمد تقي بن محمد التبريزي المامقاني (1248 هـ - 1312 هـ). هو رجل دين شيعي إيراني (آذري) من أعلام المدرسة الشيخيَّة، ويلقب المامقاني بلقب حجة الإسلام، وللمامقاني تسلُّط على الأدب والشعر الفارسي ويتخلَّص في شعره بالنيِّر. وقد ترجم له محسن الأمين ترجمة مختصرة في أعيان الشيعة وسمَّاه المولى محمد المامقاني الشيخي، واكتفى بذكر تاريخ وفاته واسم مؤلَّفين من مؤلفاته.
مؤلفاته
للمامقاني عدد من المؤلَّفات العربيَّة والفارسيَّة، ومنها:
صحيفة الأبرار في مناقب الأئمة الأطهار. أشهر مؤلَّفاته، وهو مقسم لقسمين؛ المناقب، والمعجزات، ومطبوع في خمس مجلَّدات.
مفاتيح الغيب.
اللآلي المنظومة.
كشف السبحات في تحقيق الصفات.
علم الساعة.
لمح البصر.
نصرة الحق.
رسالة حول الألف باء. يجيب فيها على كتاب الميرزا يوسف خان مستشار الدولة إليه.
ديوان شعر. أورده الطهراني في الذريعة بعنوان ديوان نير تبريزي.
آتش كده تبريزي. مثنوي فارسي.
درخوشاب. مثنوي فارسي.
تفسير آية ما خلق الجن والإنس.
مصادر
مراجع
أعيان الشيعة. محسن الأمين، طبع بيروت - لبنان، تاريخ الطبع مفقود، منشورات دار التعارف.
الذريعة إلى تصانيف الشيعة. آغا بزرگ الطهراني، طبع بيروت - لبنان، 1403 هـ / 1983 م، منشورات دار الأضواء.
أعلام شيخية
علماء دين شيعة إيرانيون
مواليد 1248 هـ
وفيات 1312 هـ
شعراء إيرانيون بالعربية
مركز بحوث التنمية الدولية

مركز بحوث التنمية الدولية هيئة حكومية كندية تعنى بالبحوث العلمية في مجال التنمية الدولية.
يتعاون مركز بحوث التنمية الدولية عن قرب مع مؤسسات البحث العلمي والتعليم العالي والهيئات الحكومية في الوطن العربي منذ عام 1971. وبعد ذلك بثلاثة أعوام أنشأ المركز مقرا دائما له في المنطقة؛ وذلك بافتتاح مكتب إقليمي في العاصمة اللبنانية بيروت، ثم انتقل مقر المكتب إلى القاهرة عام 1976، ويعد هذا المكتب مركزاً لأنشطة المركز في 10 دول تمتد من تركيا إلى المغرب. وقد دعم مركز بحوث التنمية الدولية – كندا، خلال الثلاثين عاماً الماضية ما يزيد على 550 مشروعاً للأبحاث تديرها وتشرف عليها المؤسسات والجهات البحثية في المنطقة. توجد مجموعة من المراجع الخاصة بكل دولة تقدم فكرة عامة عن أعمال المركز بها، وتتضمن هذه المراجع الأفكار الرئيسية للمشروعات.
يدعم المركز في إطار عمل برنامج واستراتيجية المؤسسات (CSPF من 2005 إلى 2010) أربعة مجالات عامة من البرامج في المنطقة وهي: إدارة البيئة والموارد الطبيعية (ENRM) وتكنولوجيا المعلومات والاتصالات من أجل التنمية (ICT4D) والإبداع والسياسة والعلوم، والسياسة الاجتماعية والاقتصادية (SEP).
مراجع
وصلات خارجية
الموقع الرسمي لمركز بحوث التنمية الدولية
تنمية عالمية
شركات تاجية اتحادية كندية
شركات مقرها في أوتاوا
مؤسسات تنمية
منظمات مقرها في أوتاوا
وزارات كندية
المنظمات التنموية الدولية
A Short History of Nearly Everything

A Short History of Nearly Everything by American-British author Bill Bryson is a popular science book that explains some areas of science, using easily accessible language that appeals more to the general public than many other books dedicated to the subject. It was one of the bestselling popular science books of 2005 in the United Kingdom, selling over 300,000 copies.
A Short History deviates from Bryson's popular travel book genre, instead describing general sciences such as chemistry, paleontology, astronomy, and particle physics. In it, he explores time from the Big Bang to the discovery of quantum mechanics, via evolution and geology.
Background
Bill Bryson wrote this book because he was dissatisfied with his scientific knowledge—that was, not much at all. He writes that science was a distant, unexplained subject at school. Textbooks and teachers alike did not ignite the passion for knowledge in him, mainly because they never delved into the whys, hows, and whens.
Contents
Bryson describes graphically and in layperson's terms the size of the universe and that of atoms and subatomic particles. He then explores the history of geology and biology and traces life from its first appearance to today's modern humans, emphasizing the development of the modern Homo sapiens. Furthermore, he discusses the possibility of the Earth being struck by a meteorite and reflects on human capabilities of spotting a meteor before it impacts the Earth, and the extensive damage that such an event would cause. He also describes some of the most recent destructive disasters of volcanic origin in the history of our planet, including Krakatoa and Yellowstone National Park.
A large part of the book is devoted to relating humorous stories about the scientists behind the research and discoveries and their sometimes eccentric behaviours. Bryson also speaks about modern scientific views on human effects on the Earth's climate and livelihood of other species, and the magnitude of natural disasters such as earthquakes, volcanoes, tsunamis, hurricanes, and the mass extinctions caused by some of these events.
An illustrated edition of the book was released in November 2005. A few editions in audiobook form are also available, including an abridged version read by the author, and at least three unabridged versions.
Awards and reviews
The book received generally favourable reviews, with reviewers citing the book as informative, well-written, and entertaining.
In 2004, this book won Bryson The Aventis Prizes for Science Books for best general science book. Bryson later donated the GBP£10,000 prize to the Great Ormond Street Hospital children's charity.
In 2005, the book won the EU Descartes Prize for science communication. It was shortlisted for the Samuel Johnson Prize for the same year.
See also
Big History
References
External links
Bill Bryson – A short history of nearly everything presentation at the Royal Society
Interview with Mariella Frostrup (BBC Radio 4)
A list of errata in A Short History of Nearly Everything
2003 non-fiction books
Books by Bill Bryson
Books about the history of science
Ephraim–Gibraltar Airport

Ephraim–Gibraltar Airport is a public use airport located one nautical mile (2 km) southwest of the central business district of Ephraim, a village in Door County, Wisconsin, United States. The airport is owned by the Village of Ephraim and the Town of Gibraltar. It was formerly known as Ephraim-Fish Creek Airport.
It is included in the Federal Aviation Administration (FAA) National Plan of Integrated Airport Systems for 2021–2025, in which it is categorized as a local general aviation facility.
History
The first runway at the airport was graded in 1945.
Facilities and aircraft
Ephraim–Gibraltar Airport covers an area of 239 acres (97 ha) at an elevation of 773 feet (236 m) above mean sea level. It has two runways: 14/32 is 2,697 by 60 feet (822 x 18 m) with an asphalt surface and 1/19 is 2,324 by 100 feet (708 x 30 m) with a turf surface.
Runway 1/19 slopes to the south such that one end of the runway can not be seen from the other. The airport provides 100LL aircraft fuel.
For the 12-month period ending August 16, 2023, the airport had 9,300 aircraft operations, an average of 25 per day, with 59% transient general aviation, 38% local general aviation and 3% air taxi. In October 2023, there were 17 aircraft based at this airport: 14 single-engine, 1 multi-engine and 2 helicopter.
Climate
See also
List of airports in Wisconsin
References
External links
Friends of Ephraim-Gibraltar Airport
Airports in Wisconsin
Buildings and structures in Door County, Wisconsin
Transportation in Door County, Wisconsin
Rachel Mwanza

Rachel Mwanza is an actress from the Democratic Republic of the Congo, best known for her performance as Komona in the 2012 film War Witch (Rebelle). Prior to being cast in the film, she was homeless and living on the streets of Kinshasa.
Early life
Mwanza was born in 1997 in Mbuji-Mayi, the third of six siblings, and spent her early childhood in Kasai Province. Her father dispatched her mother and siblings to Kinshasa when she was eight years old, promising to rejoin them later. There, the children no longer attended school, and her mother held her responsible for the family's misfortune after a false prophet stated Mwanza was a witch. He charged her mother for attempted exorcisms to rid Mwanza of witchcraft, but eventually she was thrown out on the street.
Mwanza spent several years living as a street child in Kinshasa before being cast in the Canadian film War Witch (Rebelle). She also appeared in the 2013 film Kinshasa Kids written and directed by Marc-Henri Wajnberg.
She later moved to Montreal, where she attended the high school École Lucien-Pagé and lived with the family of Anne-Marie Gélinas, the associate film producer for War Witch.
Acting career
Mwanza was cast in Rebelle after director Kim Nguyen and producers Pierre Even and Marie-Claude Poulin saw her in a documentary film on the street kids of Kinshasa. Due to her lack of education, she did not know how to read or write when she was first cast in the film; the filmmakers have since made arrangements to pay for her education and housing until she turns 18. In February 2013, she was granted a visa to allow her to attend the Academy Awards, as it was the Canadian entry for the Academy Award for Best Foreign Language Film. The visit combined attendance at the awards along with promotion work for Rebelle while she was in the United States.
Mwanza's performance in Rebelle garnered her awards including the Silver Bear for Best Actress from the Berlin Film Festival, the Tribeca Film Festival and the Vancouver Film Critics Circle in 2012, as well as the award for Best Actress at the 1st Canadian Screen Awards and Québec Cinéma.
Mwanza subsequently wrote a book titled Survivre pour voir ce jour, describing her childhood and experiences in the hope that it would both inspire young people but also bring attention to the 20,000 children who live on the streets of Kinshasha.
Her later performances have included the Belgian film Third Wedding (Troisièmes noces) in 2018.
References
Further reading
External links
1997 births
Living people
Democratic Republic of the Congo actresses
People from Kinshasa
Best Actress Genie and Canadian Screen Award winners
Silver Bear for Best Actress winners
Best Actress Jutra and Iris Award winners
SH3PXD2A

SH3PXD2A (SH3 and PX domains 2A) هوَ بروتين يُشَفر بواسطة جين SH3PXD2A في الإنسان.
الوظيفة
الأهمية السريرية
المراجع
قراءة متعمقة
جينات على كروموسوم 10
فيليب هاميلتون (شاعر)

فيليب هاميلتون (شاعر)
فيليب هاميلتون هو طالب وشاعر أمريكي، ولد في 22 يناير 1782 في ألباني في الولايات المتحدة، وتوفي في 24 نوفمبر 1801 في في الولايات المتحدة.
مراجع
أشخاص من ألباني (نيويورك)
ألكسندر هاميلتون
أمريكيون من أصل إسكتلندي
أمريكيون من أصل هولندي
خريجو جامعة كولومبيا (نيويورك)
مواليد 1782
وفيات 1801
وفيات بإطلاق النار في نيوجيرسي
Multi-index notation

Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.
Definition and basic properties
An n-dimensional multi-index is an -tuple
of non-negative integers (i.e. an element of the -dimensional set of natural numbers, denoted ).
For multi-indices and , one defines:
Componentwise sum and difference
Partial order
Sum of components (absolute value)
Factorial
Binomial coefficient
Multinomial coefficient
where .
Power
.
Higher-order partial derivative
where (see also 4-gradient). Sometimes the notation is also used.
Some applications
The multi-index notation allows the extension of many formulae from elementary calculus to the corresponding multi-variable case. Below are some examples. In all the following, (or ) and (or ).
Multinomial theorem
Multi-binomial theorem
Note that, since is a vector and is a multi-index, the expression on the left is short for .
Leibniz formula
For smooth functions and ,
Taylor series
For an analytic function in variables one has In fact, for a smooth enough function, we have the similar Taylor expansion where the last term (the remainder) depends on the exact version of Taylor's formula. For instance, for the Cauchy formula (with integral remainder), one gets
General linear partial differential operator
A formal linear -th order partial differential operator in variables is written as
Integration by parts
For smooth functions with compact support in a bounded domain one has This formula is used for the definition of distributions and weak derivatives.
An example theorem
If are multi-indices and , then
Proof
The proof follows from the power rule for the ordinary derivative; if α and β are in , then
Suppose and . Then we have that
For each in , the function only depends on . In the above, each partial differentiation therefore reduces to the corresponding ordinary differentiation . Hence, from equation, it follows that vanishes if for at least one in . If this is not the case, i.e., if as multi-indices, then
for each and the theorem follows. Q.E.D.
See also
Einstein notation
Index notation
Ricci calculus
References
Saint Raymond, Xavier (1991). Elementary Introduction to the Theory of Pseudodifferential Operators. Chap 1.1 . CRC Press.
Combinatorics
Mathematical notation
Articles containing proofs
كرملين (توضيح)

كرملين (توضيح)
الكرملين (كريمل) هي قلعة في مدينة روسية.
مبرهنة أويلر

مبرهنة أويلر
في نظرية الأعداد، مبرهنة أويلر لصاحبها ليونارد أويلر تنص على أنه إذا كان n عددا طبيعيا وa أوليا مع n، فإن a مرفع لقوة يطابق 1 قياس n:
حيث دالة أويلر أو مؤشر أويلر
في 1736، قدم أويلر إثباته لمبرهنة فيرما الصغرى، والتي قدمها فيرما دون إثبات. بعد ذلك، قدم أويلر أدلة أخرى على النظرية، وبلغت ذروتها بـ «نظرية أويلر» في ورقته البحثية عام 1763، والتي حاول فيها إيجاد أصغر أس الذي به تكون نظرية فيرما الصغرى صحيحة دائمًا.
النظرية تعد تعميماً لنظرية فيرما الصغرى، ويمكن تعميمها إلى مبرهنة كارمايكل.
يمكن استخدام المبرهنة لإيجاد البواقي لإعداد ذات قوى كبيرة ل"n" بسهولة. على سبيل المثال: لإيجاد وحدة الآحاد للعدد والذي يكافئ
إذا آحاد العدد هو 9
البرهان
لتكن <sub>نظام بواقي مصغر (mod n) ولتكن a عدداً صحيحاً أولي نسبياً مع n.
البرهان مبني على أن الضرب بـa يدوّر الباقي xi: بكلمات أخرى إذا كان فإن .
ما يعني أن المجموعات و بأخذهم (mod n) فإن لهم نفس الباقي، مايعني أن حاصل ضرب عناصر المجموعة R مساوٍ لـ aR :
وبالتالي نستطيع التخلص من xi ونحصل على مبرهنة أويلر:
راجع أيضاً
مبرهنة كارمايكل
مبرهنة فيرما الصغرى
نظرية ويلسون
المصادر
== وصلات خارجية ==
Euler's Theorem at PlanetMath
حسابيات نمطية
ليونهارت أويلر
مبرهنات في نظرية الأعداد
مقالات تحوي براهين