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研究生： 王姿丰
Wang, Zih-Fong
論文名稱： n^2+a 形式的質數
Primes of the form n^2+a
指導教授： 黃柏嶧
Huang, Po-Yi
學位類別： 碩士
Master
系所名稱： 理學院 - 數學系應用數學碩博士班
Department of Mathematics
論文出版年： 2014
畢業學年度： 102
語文別： 英文
論文頁數： 48
中文關鍵詞： 質數 、p-adic square roots 、reducible numbers 、高斯質數
外文關鍵詞： Primes, p-adic Square Roots, Reducible Numbers, Gauss prime
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    • 本文討論篩選質數形如n^2+a 的方法，尋找其與reducible numbers, p-adic square roots和高斯質數的關係，並且介紹如何估算質數形如n^2+a的數量和質數定理。

    We discuss the sieve method of primes of the form n^2+a in this thesis. Besides, we fi nd the relations about the reducible numbers, the relations about p-adic square roots
    and the relations about Gauss prime. Then we introduce how to calculate numbers of primes of the form n^2+a and the prime number theorem.

    List of Tables 8 List of Figures 9 1 Introduction 10 2 A Sieve Method for Factoring Numbers of the Form n^2+1 11 2.1 The Sieve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.2 The Program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 The primes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.4 The Reducible and Irreducible Numbers . . . . . . . . . . . . . . . . . . . 17 2.4.1 Todd's Reduction Process . . . . . . . . . . . . . . . . . . . . . . . 18 2.4.2 The Irreducible Term with Associate Primes . . . . . . . . . . . . 19 2.4.3 Arctangent Identities of  . . . . . . . . . . . . . . . . . . . . . . . 20 2.5 p-adic Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5.1 p-adic Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.5.2 Degeneracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.6 Generalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 The Conjecture of the number of Primes of the Form n^2+a 24 3.1 The Discussion of ha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 An Elementary Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . 28 4 The Prime Number Theorem 36 4.1 The Riemann Zeta Function . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.2 Newman's Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 References 47

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    [2] Hwang Chien-lih, Some observations on the method of arctangents for the calculation of , Math. Gaz., (270-278), 2004.
    [3] Hwang Chien-lih and Michael Roby Wether eld, Computing Pi, (web)http://www.machination.eclipse.co.uk/index.html, 2008.
    [4] Christopher Davis, p-adic Numbers. (web)http://www.math.umn.edu/garrett/students/reu/padic.pdf,2000.
    [5] G. H. Hardy and J. E. Littlewood, Partitio numerorum III: On the expression of a number as a sum of primes, Acta Math., (48), 1923.
    [6] J. Korevaar, The Mathematical Intelligencer, Springer Verlag, Vol 4, (108-115), 1982.
    [7] E. Landau, Aus der elementaren Zahlentheorie, Chelsea, Part IV, Chap. 6-9, 1946.
    [8] C. C. MacDuffee, An Introduction to Abstract Algebra, Wiley, New York, 1940.
    [9] Daniel Shanks, Solved and Unsolved Problems in Number Theory., Ams Chelsea Publishing, 1962.
    [10] Daniel Shanks, A Sieve Method for Factoring Numbers of the Form n2 + 1, Mathematics of Computation, (78-86), 1959.
    [11] Daniel Shanks, On the conjecture of Hardy and Littlewood concerning the number of primes of the form n2 + a, Mathematics of Computation, (321-332), 1960.
    [12] Daniel Shanks, Quadratic residues and the distribution of primes, Math. Tables Aids Comput., (272-284), 1959.
    [13] John Todd, A Problem on Arc Tangent Relations, American Math. Monthly 56, (517-528), 1949.
    [14] A. E. Western, Note on the number of primes of the form n2 +1, Proc. Cambridge Philos. Soc., (108-109), 1922.

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