What do you think?
Rate this book
Number Theory and Its History
Unusually clear, accessible introduction covers counting, properties of numbers, prime numbers, Aliquot parts, Diophantine problems, congruences, much more. Bibliography.
Hardcover
First published January 1, 1948
15 people are currently reading
175 people want to read
About the author
Øystein Ore
10 books2 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.
Ratings & Reviews
Friends & Following
Create a free account to discover what your friends think of this book!
Community Reviews
Displaying 1 - 8 of 8 reviews
November 1, 2007
This was a very fun book to work through as my summer pleasure reading one year in high school. So... er... that probably says more about me than about the book.
Anyway, if you like reading math textbooks for fun, then you totally rock! If you also happen not to know lots of number theory yet, this is a good one to try.
Anyway, if you like reading math textbooks for fun, then you totally rock! If you also happen not to know lots of number theory yet, this is a good one to try.
April 13, 2018
Very fun read. The author’s enthusiasm and curiosity are infectious, and his proof expositions are lucid and elegant; with the one exception of the chapter on primitive roots, which took a couple tries to make sense of.
March 15, 2021
A superb introduction to basic number theory - rich in historical context.
It covers the basics you would expect - primes, aliquot parts (an old fashioned term for divisors), congruences, diophantine problems. There is an emphasis on computational methods with the charming (and also old fashioned) use of 'algorism' for 'algorithm'. There is not a whiff of emphasis on computational efficiency - something that grows vastly in importance in later, current, more anguished times. Which is refreshing. ( The author does note a few times the practical impossibility of methods because the amount of computation is too vast. but doesn't stress about it. ). There are a few spots where he delves quite deeply into the intricacies - out of all proportion to the rest of the book. And I must confess I skipped over a couple of these bits. Time is short in these more desperate times. Back then, long empty days ... for the privileged few.
The book was published in 1948. The author shares his anticipation that these new fangled calculating machines developed for war might soon be released for more peaceful pursuits. And looks forward to advances in number theory from a far more extensive ability to experiment numerically - far beyond what even giant historical figures such as Gauss could ever have done. And thus it did come to pass. For a while.
He notes that no odd perfect numbers are known and that there are no known Fermat primes ( beyond the first five). But there might be. It is fascinating to note that seventy years or so later we still don't know. Though we do know that if they exist they must be truly gargantuously large. Look it up!!
Great book. Fascinating read. Charming and delightful throughout.
It covers the basics you would expect - primes, aliquot parts (an old fashioned term for divisors), congruences, diophantine problems. There is an emphasis on computational methods with the charming (and also old fashioned) use of 'algorism' for 'algorithm'. There is not a whiff of emphasis on computational efficiency - something that grows vastly in importance in later, current, more anguished times. Which is refreshing. ( The author does note a few times the practical impossibility of methods because the amount of computation is too vast. but doesn't stress about it. ). There are a few spots where he delves quite deeply into the intricacies - out of all proportion to the rest of the book. And I must confess I skipped over a couple of these bits. Time is short in these more desperate times. Back then, long empty days ... for the privileged few.
The book was published in 1948. The author shares his anticipation that these new fangled calculating machines developed for war might soon be released for more peaceful pursuits. And looks forward to advances in number theory from a far more extensive ability to experiment numerically - far beyond what even giant historical figures such as Gauss could ever have done. And thus it did come to pass. For a while.
He notes that no odd perfect numbers are known and that there are no known Fermat primes ( beyond the first five). But there might be. It is fascinating to note that seventy years or so later we still don't know. Though we do know that if they exist they must be truly gargantuously large. Look it up!!
Great book. Fascinating read. Charming and delightful throughout.
June 30, 2008
Dated...but the history is fascinating. I would only recommend this book for folks that are into advanced mathematics.
January 24, 2024
It was fine. You really need to like math in order to follow this.
July 2, 2015
It's from 1948, so it's quite out-dated; a lot of the math is describes as difficult or time-consuming can easily be done on any modern calculator. But it still offers a nice blend of math history and regular, proof-driven math, so for any higher-level math student it's worth reading.
November 13, 2016
1. counting and recording of numbers
2. properties of numbers
3. Euclid's algorithm
4. prime numbers
5. aliquot parts
6. diophantine problems
7. congruences
8. analysis of congruences
9. wilson's theorm
10. euler's theorm
11. fermat's theorm
2. properties of numbers
3. Euclid's algorithm
4. prime numbers
5. aliquot parts
6. diophantine problems
7. congruences
8. analysis of congruences
9. wilson's theorm
10. euler's theorm
11. fermat's theorm
April 4, 2012
Fun introduction to number theory with history thrown in. Good examples and helps clear some confusion about some math concepts.
Displaying 1 - 8 of 8 reviews