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Early Study in Which Leonhard Euler Made Use of the Mathematical Analysis to the Research of Number Theory(PDF)

《南京师大学报（自然科学版）》[ISSN:1001-4616/CN:32-1239/N]

Issue:
2007年03期
Page:
34-38
Research Field:
数学
Publishing date:

Info

Title:
Early Study in Which Leonhard Euler Made Use of the Mathematical Analysis to the Research of Number Theory
Author(s):
Tang Zhihua1; Xu Liquan2; HSU Leetsch Charles3
1.Nanjing Higher Normal Preschool,Nanjing 210004,China
2. Wuxi Teaching and Research Center,Wuxi 214001, China)
3. Department ofMathematics, Dalian University of Technology, Dalan 116024, China
Keywords:
Leonhard Euler;  number theory; Bernoulli numbers;  power series expansion;  Eular p roduct
PACS:
O156
DOI:
-
Abstract:
Here p resented is a brief introduction of Euler’s study. There are many exciting formulas and theorems such as Euler’s infinite summation formula about recip rocal sum with powers of the natural num bers and Euler’s infinite p roduct rep resentation. Another examp le is Euler’s thinking and p roving about the p roof of Fermat’s four‘square theorem, yielding the arithmetical function, the partition function, and p rime ideal. These concep tions, theorems, and formulas were all first discovered accurately by Euler’s demonstrations. Eulerwas extraordinary at converting a p roblem of number theory into mathematical anal ysis. In fact, Euler’s ideas have become more generalized. These facts are enough to p rove that he had extensive and deep knowledge of his subject. Finally, we quoted a few famous examp les of power series, and the law of quadratic recip rocity which Euler found. They are all part of our p recious legacy in the li brary of number theory from Euler.

References:

[ 1 ] 　ScharlauW, Hans Opolka. From Fermat to Minkowski [ C ] / / Lectures on Theory of Numbers and its Historical Development. New York: Sp ringer2Verlag, 1985: 13-31.
[ 2 ] 　KlineM. Mathematical Thought From Ancient toModern Times[M ]. New York: Oxford University Press, 1990.
[ 3 ] 　华罗庚, 王元. 数论在近似分析中的应用[M ]. 北京:科学出版社, 1978: 128-129.
[ 4 ] 　潘承洞,潘承彪. 初等代数数论[M ]. 济南:山东大学出版社, 1991: 112-117.
[ 5 ] 　徐利治. 论数学方法学[M ]. 济南: 山东教育出版社, 2001: 598-600.
[ 6 ] 　徐沥泉. 首屈一指的数学大师———纪念欧拉诞生300周年[ J ]. 自然杂志, 2007, 29 (3) : 1832186

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Last Update: 2013-05-05