参考文献

Adleman, Leonard M. 1980. “On Distinguishing Prime Numbers from Composite Numbers.” In 21st Annual Symposium on Foundations of Computer Science (Sfcs 1980), 387–406. IEEE. https://doi.org/10.1109/sfcs.1980.28.

Agrawal, Manindra, Neeraj Kayal, and Nitin Saxena. 2004. “PRIMES Is in p.” Annals of Mathematics 160 (2): 781–93.

Atkin, A. O. L., and F. Morain. 1993. “Elliptic Curves and Primality Proving.” Mathematics of Computation 61: 29–68.

Bailey, D. H., P. B. Borwein, and S. Plouffe. 1996. “On the Computation of the n’th Decimal Digit of Various Transcendental Numbers.” Mathematics of Computation, 1–2. http://lacim.uqam.ca/~plouffe/Simon/articlepi.html.

Baillie, Robert, and Samuel S. Wagstaff Jr. 1980. “Lucas Pseudoprimes.” Mathematics of Computation 35 (152): 1391–1417.

Beeler, M., R. W. Gosper, and R. Schroeppel. 1972. HAKMEM. Cambridge, MA: MIT Artificial Intelligence Laboratory, Memo AIM-239.

Bellard, F. 1997. “A New Formula to Compute the n’th Binary Digit of Pi.” Bellard’s Website, 1–2. http://fabrice.bellard.free.fr/pi/pi_bin/pi_bin.html.

Berndt, B. C. 1994. Ramanujan’s Notebooks, Part IV. New York: Springer-Verlag.

Borwein, J. M., and P. B. Borwein. 1987. Pi and the AGM. Addison-Wesley Longman Publishing Co., Inc.

———. 1988. “More Ramanujan-Type Series for 1/PI.” In Ramanujan Revisited: Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, June 1-5, 1987, edited by G. E. Andrews, B. C. Berndt, and R. A. Rankin, 359–74. New York: Academic Press.

Borwein, J. M., P. B. Borwein, and D. H. Bailey. 1989. “Ramanujan, Modular Equations, and Approximations to Pi, or How to Compute One Billion Digits of Pi.” Amer. Math. Monthly 96, 201–19.

Bos, Jurjen, and Matthijs Coster. 1990. “Addition Chain Heuristics.” In Advances in Cryptology - Proceedings of Crypto ’89, 435:400–407. Springer-Verlag.

Brent, R. P. 1975. “Multiple-Precision Zero-Finding Methods and the Complexity of Elementary Function Evaluation.” Analytic Computational Complexity, 151–76. http://wwwmaths.anu.edu.au/~brent/pub/pub028.html.

Brent, Richard P. 1980. “An Improved Monte Carlo Factorization Algorithm.” BIT Numerical Mathematics 20 (2): 176–94.

Bruce, J. W. 1993. “A Really Trivial Proof of the Lucas-Lehmer Test.” The American Mathematical Monthly 100 (4): 370–71. https://doi.org/10.1080/00029890.1993.11990414.

Burnikel, C., J. Ziegler, I. Stadtwald, and D. Saarbrucken. 1998. “Fast Recursive Division.”

Chudnovsky, David V., and Gregory V. Chudnovsky. 1987. “Approximations and Complex Multiplication According to Ramanujan.” In Ramanujan Revisited: Proceedings of the Centenary Conference, University of Illinois at Urbana-Champaign, June 1-5, 1987, edited by G. E. Andrews, B. C. Berndt, and R. A. Rankin, 375–472. Boston, MA: Academic Press.

———. 1990. “Computer Algebra in the Service of Mathematical Physics and Number Theory.” Computers and Mathematics 09: 232.

Cohen, Henri. 1993. A Course in Computational Algebraic Number Theory. GTM 138. Springer Verlag.

Cohen, H., and H. W. Lenstra. 1984. “Primality Testing and Jacobi Sums.” Mathematics of Computation 42 (165): 297–330. https://doi.org/10.1090/s0025-5718-1984-0726006-x.

Corman著，潘金贵等译, H. 2006. 算法导论(原书第2版). 机械工业出版社.

Deléglise, M., and J. Rivat. 1996a. “Computing the Summation of the Möbius Function.” Experimental Mathematics 5 (4): 291–95.

———. 1996b. “Computing π(x): The Meissel, Lehmer, Lagarias, Miller, Odlyzko Method.” Mathematics of Computation 65 (213): 235–45.

Dixon, John D. 1981. “Asymptotically Fast Factorization of Integers.” Mathematics of Computation 36 (153): 255–60.

Gathen, Joachim von zur, and Jürgen Gerhard. 2002. Modern Computer Algebra. 2nd ed. Cambridge University Press.

Goldwasser, S., and J. Kilian. 1986. “Almost All Primes Can Be Quickly Certified.” In Proceedings of the Eighteenth Annual ACM Symposium on Theory of Computing, 316–29.

Gordon, Daniel M. 1998. “A Survey of Fast Exponentiation Methods.” Journal of Algorithms 27 (1): 129–46.

Gourdon, X. 2001. “Computation of  pi(x) : Improvements to the Meissel, Lehmer, Lagarias, Miller, Odlyzko, Deleglise and Rivat Method.”

Gower, Jason E., and Samuel S. Wagstaff. 2008. “Square Form Factorization.” Mathematics of Computation 77 (261): 551–88. https://doi.org/10.1090/s0025-5718-07-02010-8.

Gutierrez, C., and M. Monsalve. 2005. “Fast Inverse for Big Numbers: Picarte’s Iteration.”

H．J．Nussbaumer. 1982. Fast Fourier Transform and Convolution Algorithms. Berlin: Springer-Verlag.

H．J．Nussbaumer, and 胡光锐译. 1984. 快速付里叶变换和卷积算法. 上海: 上海科学技术文献出版社.

Hong, Seong Min, Sang Yeop Oh, and Hyun Soo Yoon. 1996. “New Modular Multiplication Algorithms for Fast Modular Exponentiation.” In Advances in Cryptology—Proceedings of Eurocrypt ’96, 1070:166–77. Lecture Notes in Computer Science. Springer Berlin / Heidelberg.

Hwang, C. L. 1997. “More Machin-Type Identities.” Math. Gaz. 81: 120–21.

Jebelean, T. 1993. “A Generalization of the Binary GCD Algorithm.” Proceedings of the 1993 International Symposium on Symbolic and Algebraic Computation, 111–16.

Kanada, Yasumasa. 2005. “Current Published World Record of Pi Calculation.”

Karatsuba, A., and P. Ofman. 1962. “Multiplication of Many-Digital Numbers by Automatic Computers.” Proceedings of the USSR Academy of Sciences, no. 145: 293–94.

Knuth, Donald E. 1997. The Art of Computer Programming, Volume 2: Seminumerical Algorithms. 3rd ed. Boston, MA, USA: Addison-Wesley Longman Publishing Co., Inc.

Lagaria, J. C., V. S. Miller, and A. M. Odlyzko. 1985. “Computing π(x): The Meissel-Lehmer Method.” Mathematics of Computation 44 (170): 537–60.

Lagarias, J. C., and A. M. Odlyzko. 1987. “Computing π(x): An Analytic Method.” Journal of Algorithms 8 (2): 173–91.

Lehmer, D. H. 1930. “An Extended Theory of Lucas’ Functions.” The Annals of Mathematics 31 (3): 419. https://doi.org/10.2307/1968235.

———. 1932. “An Inversive Algorithm.” Bulletin of the American Mathematical Society 38 (10): 693–94.

———. 1938. “On Arccotangent Relations for π.” Amer. Math. Monthly 45: 657–64.

Lenstra, A. K., H. W. Lenstra Jr., M. S. Manasse, and J. M. Pollard. 1993. “The Factorization of the Ninth Fermat Number.” Mathematics of Computation 61 (203): 319–49.

Lenstra, H. W., Jr. 1987. “Factoring Integers with Elliptic Curves.” The Annals of Mathematics 126 (3): 649–73.

M. Mendès France, G. Tenenbaum, and 姚家燕译. 2007. 素数论. 研究生数学丛书. 北京: 清华大学出版社.

Martin, M. 2004. “Re: Baillie-PSW - Which Variant Is Correct?”

Montgomery, Peter L. 1985. “Modular Multiplication Without Trial Division.” Mathematics of Computation 44 (170): 519–21.

———. 1987. “Speeding the Pollard and Elliptic Curve Methods of Factorization.” Mathematics of Computation 48 (177): 243–64.

Morrison, Michael A., and John Brillhart. 1975. “A Method of Factoring and the Factorization of F₇.” Mathematics of Computation 29 (129): 183–205.

Pagliarulo, Steve. 2007. “Stu’s Pi Page: Other Constants.” https://web.archive.org/web/20080825030820/http://home.istar.ca/~lyster/otherconstants.html.

Pollard, J. M. 1974. “Theorems on Factorization and Primality Testing.” Mathematical Proceedings of the Cambridge Philosophical Society 76 (3): 521–28.

———. 1975. “A Monte Carlo Method for Factorization.” BIT Numerical Mathematics 15 (3): 331–34.

Pomerance, Carl. 1982. “Analysis and Comparison of Some Integer Factoring Algorithms.” In Computational Methods in Number Theory, 89–139. Mathematisch Centrum.

———. 1984. “Are There Counterexamples to the Baillie-PSW Primality Test?”

Pomerance, Carl, J. L. Selfridge, and Samuel S. Wagstaff. 1980. “The Pseudoprimes to 25 ⋅ 10⁹.” Mathematics of Computation 35 (151): 1003–26. https://doi.org/10.1090/s0025-5718-1980-0572872-7.

Rabin, Michael O. 1980. “Probabilistic Algorithm for Testing Primality.” Journal of Number Theory 12 (1): 128–38. https://doi.org/10.1016/0022-314x(80)90084-0.

Riesel, Hans. 1985. Prime Numbers and Computer Methods for Factorization. PM 57. Boston; Basel; Stuttgart: Birkhäuser.

Schoenhage, A., and V. Strassen. 1971. “Schnelle Multiplikation Groeer Zahlen.” Computing, no. 7: 281–92.

Sedjelmaci, Sidi Mohamed. 2007. “Jebelean–Weber’s Algorithm Without Spurious Factors.” Information Processing Letters 102 (6): 247–52.

Selfridge, J. L., and A. Hurwitz. 1964. “Fermat Numbers and Mersenne Numbers.” Mathematics of Computation 18: 146–48.

Solovay, R., and V. Strassen. 1977. “A Fast Monte-Carlo Test for Primality.” SIAM Journal on Computing 6 (1): 84–85. https://doi.org/10.1137/0206006.

Sorenson, Jonathan. 1994. “Two Fast GCD Algorithms.” Journal of Algorithms 16 (1): 110–44.

The GMP team. 2008. The GNU Multiple Precision Arithmetic Library. 4.2.3 ed.

———. n.d. “The GNU MP Bignum Library.”

Weber, Kenneth. 1995. “The Accelerated Integer GCD Algorithm.” ACM Transactions on Mathematical Software 21 (March): 111–22.

Weisstein, Eric W. n.d.a. “Horner’s Rule.” MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/HornersRule.html.

———. n.d.b. “Machin-Like Formulas.” MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/Machin-LikeFormulas.html.

———. n.d.c. “Pi Formulas.” MathWorld–A Wolfram Web Resource. https://mathworld.wolfram.com/PiFormulas.html.

Williams, H. C. 1982. “A p + 1 Method of Factoring.” Mathematics of Computation 39 (159): 225–34.

冯克勤. 2003. 初等数论及应用. 走进数学新课程丛书. 北京: 北京师范大学出版社.

李庆扬, 王能超, and 易大义. 2001. 数值分析. 4th ed. 北京: 清华大学出版社.

王东明, 夏壁灿, and 李子明. 2007. 计算机代数（第二版）. 2nd ed. 计算科学丛书. 北京: 清华大学出版社.

王竹溪, and 郭敦仁. 1965. 特殊函数概论. 1st ed. 北京: 科学出版社.

聂灵沼, and 丁石孙. 2000. 代数学引论（第2版）. 2nd ed. 面向21世纪课程教材（数学类）. 北京: 高等教育出版社.

裴定一, and 祝跃飞. 2002. 算法数论. 中国科学院研究生教学丛书. 北京: 科学出版社.

颜松远. 2008. 计算数论. 2nd ed. 北京: 清华大学出版社.