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Anneli Lax New Mathematical Library
Приглашение в теорию чисел
Книга известного норвежского математика О. Оре раскрывает красоту математики на примере одного из ее старейших разделов ? теории чисел. Изложение основ теории чисел в книге во многом нетрадиционно. Наряду с теорией сравнений, сведениями о системах счисления, в ней содержатся рассказы о магических квадратах, о решении арифметических ребусов и т. д. Большим достоинством книги является то, что автор при каждом удобном случае указывает на возможности практического применения изложенных результатов, а также знакомит читателя с современным состоянием теории чисел и задачами, еще не получившими окончательного решения.
- GenresMathematics
128 pages, Paperback
First published June 1, 1967
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48 people want to read
About the author
Øystein Ore
22 books3 followersOystein Ore (7 October 1899 – 13 August 1968) was a Norwegian mathematician.
Ore graduated from the University of Oslo in 1922, with a Cand.Scient. degree in mathematics. In 1924, the University of Oslo awarded him the Ph.D. for a thesis titled Zur Theorie der algebraischen Körper, supervised by Thoralf Skolem. Ore also studied at Göttingen University, where he learned Emmy Noether's new approach to abstract algebra. He was also a fellow at the Mittag-Leffler Institute in Sweden, and spent some time at the University of Paris. In 1925, he was appointed research assistant at the University of Oslo.
Yale University’s James Pierpont went to Europe in 1926 to recruit research mathematicians. In 1927, Yale hired Ore as an assistant professor of mathematics, promoted him to associate professor in 1928, then to full professor in 1929. In 1931, he became a Sterling Professor (Yale's highest academic rank), a position he held until he retired in 1968.
Ore was an AMS Colloquium Lecturer in 1941 and plenary speaker at the International Congress of Mathematicians in 1936 in Oslo. He was also elected to the American Academy of Arts and Sciences and the Oslo Academy of Science. He was a founder of the Econometric Society.
Ore visited Norway nearly every summer. During World War II, he was active in the "American Relief for Norway" and "Free Norway" movements. In gratitude for the services rendered to his native country during the war, he was decorated in 1947 with the Order of St. Olav.
In 1930, Ore married Gudrun Lundevall. They had two children. Ore had a passion for painting and sculpture, collected ancient maps, and spoke several languages.
Ore is known for his work in ring theory, Galois connections, and most of all, graph theory. His early work was on algebraic number fields, how to decompose the ideal generated by a prime number into prime ideals. He then worked on noncommutative rings, proving his celebrated theorem on embedding a domain into a division ring. He then examined polynomial rings over skew fields, and attempted to extend his work on factorisation to non-commutative rings.
As a teacher, he is notable for teaching mathematics to two doctoral students who would make vital contributions to science and mathematics : Grace Hopper, who would eventually become a United States rear admiral and computer scientist who was vital to the development of the first computers, and Marshall Hall, Jr., an American mathematician who did important research in group theory and combinatorics.
In 1930 the Collected Works of Richard Dedekind were published in three volumes, jointly edited by Ore and Emmy Noether. He then turned his attention to lattice theory becoming, together with Garrett Birkhoff, one of the two founders of American expertise in the subject. Ore's early work on lattice theory led him to the study of equivalence and closure relations, Galois connections, and finally to graph theory, which occupied him to the end of his life.
Ore had a lively interest in the history of mathematics, and was an unusually able author of books for laypeople, such as his biographies of Cardano and Niels Henrik Abel.
Ore graduated from the University of Oslo in 1922, with a Cand.Scient. degree in mathematics. In 1924, the University of Oslo awarded him the Ph.D. for a thesis titled Zur Theorie der algebraischen Körper, supervised by Thoralf Skolem. Ore also studied at Göttingen University, where he learned Emmy Noether's new approach to abstract algebra. He was also a fellow at the Mittag-Leffler Institute in Sweden, and spent some time at the University of Paris. In 1925, he was appointed research assistant at the University of Oslo.
Yale University’s James Pierpont went to Europe in 1926 to recruit research mathematicians. In 1927, Yale hired Ore as an assistant professor of mathematics, promoted him to associate professor in 1928, then to full professor in 1929. In 1931, he became a Sterling Professor (Yale's highest academic rank), a position he held until he retired in 1968.
Ore was an AMS Colloquium Lecturer in 1941 and plenary speaker at the International Congress of Mathematicians in 1936 in Oslo. He was also elected to the American Academy of Arts and Sciences and the Oslo Academy of Science. He was a founder of the Econometric Society.
Ore visited Norway nearly every summer. During World War II, he was active in the "American Relief for Norway" and "Free Norway" movements. In gratitude for the services rendered to his native country during the war, he was decorated in 1947 with the Order of St. Olav.
In 1930, Ore married Gudrun Lundevall. They had two children. Ore had a passion for painting and sculpture, collected ancient maps, and spoke several languages.
Ore is known for his work in ring theory, Galois connections, and most of all, graph theory. His early work was on algebraic number fields, how to decompose the ideal generated by a prime number into prime ideals. He then worked on noncommutative rings, proving his celebrated theorem on embedding a domain into a division ring. He then examined polynomial rings over skew fields, and attempted to extend his work on factorisation to non-commutative rings.
As a teacher, he is notable for teaching mathematics to two doctoral students who would make vital contributions to science and mathematics : Grace Hopper, who would eventually become a United States rear admiral and computer scientist who was vital to the development of the first computers, and Marshall Hall, Jr., an American mathematician who did important research in group theory and combinatorics.
In 1930 the Collected Works of Richard Dedekind were published in three volumes, jointly edited by Ore and Emmy Noether. He then turned his attention to lattice theory becoming, together with Garrett Birkhoff, one of the two founders of American expertise in the subject. Ore's early work on lattice theory led him to the study of equivalence and closure relations, Galois connections, and finally to graph theory, which occupied him to the end of his life.
Ore had a lively interest in the history of mathematics, and was an unusually able author of books for laypeople, such as his biographies of Cardano and Niels Henrik Abel.
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Displaying 1 - 2 of 2 reviews
July 14, 2017
It's a good book, but ta not quite as amazing as his graph theory book.
November 21, 2012
A charming and very readable introduction to number theory perfectly pitched for readers with intermediate high school knowledge of mathematics (a basic knowledge of divisibility and of Pythagoras´ theorem is probably sufficient to carry the reader through). The editors state that: "This book is one of a series written by professional mathematicians in order to make some important mathematical ideas interesting and readable to a large audience of high school students and laymen." Measured by this goal, I believe the book is wonderfully successful and hence the five stars in my rating.
The text niftily and succintly weaves some interesting history, beguiling mathematical problems and, in general, clear proofs of key theorems and assertions together with a smattering of problems to induce the reader to work out and understand the a well chosen set of related mathematical concepts. The books is an entertaining stroll which leads up to the solutions of the Pythagorean equation (for which integer values of the three sides are there right-angled triangles?) and to Fermat´s congruence (also known as the little Fermat´s theory) but pointing out important features on the way such as magic squares, Mersenne and Fermat primes, perfect and amicable numbers, numeration systems, the algebra of congruences and its application to checks on computations, finding out the day of the week any past or future date falls and how to build round-robin tournament schedules, amongst others. Written in 1967, it is a bit dated as regards some aspects of the field such as, understandably, the largest primes calculated using computers and I´m sure that, were Professor Ore (1899-1968) alive today and updating the book, he would carefully add a little something about the applications of number theory to cryptography. However, this does not detract from the merits of this delightful gem of a book.
It is worth recalling the editors´ key recommendation, the reader "should keep in mind that a book on mathematics cannot be read quickly. Nor must he expect to understand all parts of the book on first reading. He should feel free to skip complicated parts and return to them later; often an argument will be clarified by a subsequent remark". Finally they are perfectly right in stating that "The best way to learn mathematics is to do mathematics" and to that extent the problems set by the author are satisfactory.
The text niftily and succintly weaves some interesting history, beguiling mathematical problems and, in general, clear proofs of key theorems and assertions together with a smattering of problems to induce the reader to work out and understand the a well chosen set of related mathematical concepts. The books is an entertaining stroll which leads up to the solutions of the Pythagorean equation (for which integer values of the three sides are there right-angled triangles?) and to Fermat´s congruence (also known as the little Fermat´s theory) but pointing out important features on the way such as magic squares, Mersenne and Fermat primes, perfect and amicable numbers, numeration systems, the algebra of congruences and its application to checks on computations, finding out the day of the week any past or future date falls and how to build round-robin tournament schedules, amongst others. Written in 1967, it is a bit dated as regards some aspects of the field such as, understandably, the largest primes calculated using computers and I´m sure that, were Professor Ore (1899-1968) alive today and updating the book, he would carefully add a little something about the applications of number theory to cryptography. However, this does not detract from the merits of this delightful gem of a book.
It is worth recalling the editors´ key recommendation, the reader "should keep in mind that a book on mathematics cannot be read quickly. Nor must he expect to understand all parts of the book on first reading. He should feel free to skip complicated parts and return to them later; often an argument will be clarified by a subsequent remark". Finally they are perfectly right in stating that "The best way to learn mathematics is to do mathematics" and to that extent the problems set by the author are satisfactory.
Displaying 1 - 2 of 2 reviews