The investigation of three problems, perfect numbers, periodic decimals, and Pythagorean numbers, has given rise to much of elementary number theory. In this book, Daniel Shanks, past editor of Mathematics of Computation, shows how each result leads to further results and conjectures. The outcome is a most exciting and unusual treatment. This edition contains a new chapter presenting research done between 1962 and 1978, emphasizing results that were achieved with the help of computers.
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The flavor of ... mathematics is in this book--this is its claim to distinction. --Science
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23 of 23 people found the following review helpful:
5.0 out of 5 stars
Different Approch to Number theory,
By A Customer
This review is from: Solved and Unsolved Problems in Number Theory (CHEL/297) (Hardcover)
The author develops the premise that modern number theory evolved from the ancient Greek preoccupation with two mathematical problems. Searching for the esoteric Perfect Numbers (i.e., whole numbers whose proper divisors sum to the number itself 1+2+3=6) and Diophantine Equations (i.e., finding integral solutions to certain algebraic equations, for example, z^2 = x^2 + y^2). The author calls the later Pythagorianism. The book does a good job of showing how Fermat's Little Theorem, Euler's generalization, and the famous Law of Quadradic Reciprocity developed out of the search for Mersenne Primes, and consequently Perfect Numbers. Again, it is interesting to see how Pythagorianism led to the development of algebraic numbers and eventually to the solution of Fermats "Big" Theorem. Along the way the author elaborates on some of the still unresolved conjectures within number theory. The writing can be a little "dense" at times, so that some parts require a second reading. Overall the book is enjoyable to read and you will gain some insight that won't be gleaned from more standard texts.
1 of 29 people found the following review helpful:
5.0 out of 5 stars
pankajmath,
By pankaj gupta (india) - See all my reviews
This review is from: Solved and Unsolved Problems in Number Theory (CHEL/297) (Hardcover)
an alternate proof to show that combination coeffiecientC(p^n,r)is divisible by p; where p a prime n & r +ve integers. idea first we write the expression of c(p^n,r) & then we can conclude from the expression that this is divisible by p (how ?).
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Inside This Book
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First Sentence:
Many of the basic theorems of number theory -stem from two problems investigated by the Greeksthe problem of perfect numbers and that of Pythagorean numbers. Read the first page Key Phrases - Capitalized Phrases (CAPs): (learn more)
Euler's Criterion, Fermat's Theorem, Reciprocity Law, Euclid's Algorithm, Pythagorean Theorem, Fermat's Last Theorem, Gauss's Criterion, Box Principle, Euclid's Theorem, Euler's Theorem, Lucas Criterion, Old Pythagoreanism, Riemann Hypothesis, Thue's Theorem, Dirichlet's Theorem, New Pythagorean, Waring's Problem New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Many of the basic theorems of number theory -stem from two problems investigated by the Greeksthe problem of perfect numbers and that of Pythagorean numbers. Read the first page Key Phrases - Capitalized Phrases (CAPs): (learn more)
Euler's Criterion, Fermat's Theorem, Reciprocity Law, Euclid's Algorithm, Pythagorean Theorem, Fermat's Last Theorem, Gauss's Criterion, Box Principle, Euclid's Theorem, Euler's Theorem, Lucas Criterion, Old Pythagoreanism, Riemann Hypothesis, Thue's Theorem, Dirichlet's Theorem, New Pythagorean, Waring's Problem New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 12
books:
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36
books
cite this book:
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- Number Theory by Helmut Hasse in Back Matter
- The Psychology of Invention in the Mathematical Field by Jacques Hadamard in Back Matter
- Theory of Numbers by George B. Mathews in Back Matter
- History of the Theory of Numbers (AMS Chelsea Publishing) Volumes I; II; III. THREE VOLUME SET (Vol. 2) by Leonard Eugene Dickson in Back Matter
- Algebraic Number Theory (Graduate Texts in Mathematics) by Serge Lang in Back Matter
See all 12 books this book cites
- Fermat's Last Theorem for Amateurs by Paulo Ribenboim on page 10, page 142, and Back Matter
- A Course in Number Theory and Cryptography (Graduate Texts in Mathematics) by Neal Koblitz on page 30, and page 53
- The Penguin Book of Curious and Interesting Numbers: Revised Edition (Penguin Press Science) by David Wells on page 62, and page 89
- Philosophy of Mathematics: An Introduction to a World of Proofs and Pictures (Philosophical Issues in Science) by James Robert Brown in Back Matter
- Introduction to Modern Number Theory: Fundamental Problems, Ideas and Theories (Encyclopaedia of Mathematical Sciences) by Yu. I. Manin in Back Matter
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