Number Theory: An Introduction via the Distribution of Primes
Jun 2007 · Springer Science & Business Media
2.0star
1 reviewreport
Ebook
342
Pages
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About this ebook
Number theory is fascinating. Results about numbers often appear magical, both in theirstatementsandintheeleganceoftheirproofs. Nowhereisthismoreevidentthan inresultsaboutthesetofprimenumbers. Theprimenumbertheorem,whichgivesthe asymptotic density of the prime numbers, is often cited as the most surprising result in all of mathematics. It certainly is the result that is hardest to justify intuitively. The prime numbers form the cornerstone of the theory of numbers. Many, if not most, results in number theory proceed by considering the case of primes and then pasting the result together for all integers using the fundamental theorem of arithmetic. The purpose of this book is to give an introduction and overview of number theory based on the central theme of the sequence of primes. The richness of this somewhat unique approach becomes clear once one realizes how much number theoryandmathematicsingeneralareneededinordertolearnandtrulyunderstandthe prime numbers. Our approach provides a solid background in the standard material as well as presenting an overview of the whole discipline. All the essential topics are covered: fundamental theorem of arithmetic, theory of congruences, quadratic reciprocity, arithmetic functions, the distribution of primes. In addition, there are ?rm introductions to analytic number theory, primality testing and cryptography, and algebraic number theory as well as many interesting side topics. Full treatments and proofs are given to both Dirichlet’s theorem and the prime number theorem. There is acompleteexplanationofthenewAKSalgorithm,whichshowsthatprimalitytesting is of polynomial time. In algebraic number theory there is a complete presentation of primes and prime factorizations in algebraic number ?elds.
Ratings and reviews
2.0
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A Google user
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August 23, 2009
I was disappointed by this book overall, but especially the proofs. Many were jumpy and hard to follow, not clearly explaining what was going on. Some left out important truth conditions, and several had fatal typos that made the math impossible to follow. The whole text seemed like it had been rushed to the printer without thorough editing or review. The 2nd edition will hopefully be better.
A Google user
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good phi
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