| Diophantine Equations and Inequalities in Algebraic Number Fields Softcover Repri Edition Contributor(s): Wang, Yuan (Author) |
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ISBN: 3642634893 ISBN-13: 9783642634895 Publisher: Springer OUR PRICE: $52.24 Product Type: Paperback - Other Formats Published: October 2012 |
| Additional Information |
| BISAC Categories: - Mathematics | Number Theory |
| Dewey: 512.72 |
| Physical Information: 0.41" H x 6.69" W x 9.61" (0.70 lbs) 170 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: The circle method has its genesis in a paper of Hardy and Ramanujan (see Hardy 1])in 1918concernedwiththepartitionfunction andtheproblemofrep- resenting numbers as sums ofsquares. Later, in a series of papers beginning in 1920entitled "some problems of'partitio numerorum''', Hardy and Littlewood (see Hardy 1]) created and developed systematically a new analytic method, the circle method in additive number theory. The most famous problems in ad- ditive number theory, namely Waring's problem and Goldbach's problem, are treated in their papers. The circle method is also called the Hardy-Littlewood method. Waring's problem may be described as follows: For every integer k 2 2, there is a number s= s( k) such that every positive integer N is representable as (1) where Xi arenon-negative integers. This assertion wasfirst proved by Hilbert 1] in 1909. Using their powerful circle method, Hardy and Littlewood obtained a deeper result on Waring's problem. They established an asymptotic formula for rs(N), the number of representations of N in the form (1), namely k 1 provided that 8 2 (k - 2)2 - +5. Here |