Prime Numbers

Prime Numbers Prime Numbers

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548 REFERENCES [Apostol 1986] T. Apostol. Introduction to Analytic Number Theory, 3rd printing. Springer–Verlag, 1986. [Arazi 1994] B. Arazi. On primality testing using purely divisionless operations. The Computer Journal, 37:219–222, 1994. [Archibald 1949] R. Archibald. Outline of the history of mathematics. Amer. Math. Monthly, 56, 1949. The second Herbert Ellsworth Slaught Memorial Paper: supplement to no. 1 issue, 114 pp. [Ares and Castro 2004] S. Ares and M. Castro. Hidden structure in the randomness in the prime number sequence. Condensed Matter Abstracts, 2004. http://arxiv.org/abs/cond-mat/0310148. [Arney and Bender 1982] J. Arney and E. Bender. Random mappings with constraints on coalescence and number of origins. Pacific J. Math. 103:269–294, 1982. [Artjuhov 1966/67] M. Artjuhov. Certain criteria for the primality of numbers connected with the little Fermat theorem (Russian). Acta Arith., 12:355–364, 1966/67. [Ashworth and Lyne 1988] M. Ashworth and A. Lyne. A segmented FFT algorithm for vector computers. Parallel Computing, 6:217–224, 1988. [Atkin 1986] A. Atkin. Schoof’s algorithm. Unpublished manuscript, 1986. [Atkin 1988] A. Atkin. The number of points on an elliptic curve modulo a prime (i). Unpublished manuscript, 1988. [Atkin 1992] A. Atkin. The number of points on an elliptic curve modulo a prime (ii). Unpublished manuscript, 1992. [Atkin and Bernstein 2004] A. Atkin and D. Bernstein. Prime sieves using binary quadratic forms. Math. Comp., 73:1023–1030, 2004. [Atkin and Morain 1993a] A. Atkin and F. Morain. Finding suitable curves for the elliptic curve method of factorization. Math. Comp., 60:399–405, 1993. [Atkin and Morain 1993b] A. Atkin and F. Morain. Elliptic curves and primality proving. Math. Comp., 61:29–68, 1993. [Bach 1985] E. Bach. Analytic Methods in the Analysis and Design of Number-Theoretic Algorithms. A 1984 ACM Distinguished Dissertation. The MIT Press, 1985. [Bach 1990] E. Bach. Explicit bounds for primality testing and related problems. Math. Comp., pages 355–380, 1990. [Bach 1991] E. Bach. Toward a theory of Pollard’s rho method. Inform. and Comput., 90:139–155, 1991. [Bach 1997a] E. Bach. The complexity of number-theoretic constants. Inform. Process. Lett., 62:145–152, 1997. [Bach 1997b] E. Bach. Comments on search procedures for primitive roots. Math. Comp., 66(220):1719–1727, 1997.

REFERENCES 549 [Bach and Shallit 1996] E. Bach and J. Shallit. Algorithmic Number Theory, volume I. MIT Press, 1996. [Baillie and Wagstaff 1980] R. Baillie and S. Wagstaff, Jr. Lucas pseudoprimes. Math. Comp., 35:1391–1417, 1980. [Bailey 1990] D. Bailey. FFTs in external or hierarchical memory. J. Supercomp., 4:23–35, 1990. [Bailey and Crandall 2001] D. Bailey and R. Crandall. On the random character of fundamental constant expansions, Experiment. Math., 10:175–190, 2001. [Bailey and Crandall 2002] D. Bailey and R. Crandall. Random generators and normal numbers. Experiment. Math., 11:527–546, 2002. [Bailey et al. 2003] D. Bailey, J. Borwein, R. Crandall, and C. Pomerance. On the binary expansions of algebraic numbers. Journal de Theorie des Nombres, Bordeau (to appear), 2003. [Balasubramanian and Nagaraj 1997] R. Balasubramanian and S. Nagaraj. Density of Carmichael numbers with three prime factors. Math. Comp., 66:1705–1708, 1997. [Balazard et al. 1999] M. Balazard, E. Saias, and M. Yor. Notes sur la fonction ζ de Riemann. II. Adv. Math., 143:284–287, 1999. [Balog 1989] A. Balog. On a variant of the Piatetski-Shapiro prime number theorem. In Groupe de travail en théorie analytique et élementaire des nombres, 1987–1988, volume 89-01 of Publ. Math. Orsay, pages 3–11. Univ. Paris XI, Orsay, 1989. [Barrett 1987] P. Barrett. Implementing the Rivest Shamir and Adleman public key encryption algorithm on a standard digital signal processor. In A. Odlyzko, editor, Advances in Cryptology, Proc. Crypto ’86, volume 263 of Lecture Notes in Computer Science, pages 311–323. Springer–Verlag, 1987. [Bateman et al. 1989] P. Bateman, J. Selfridge, and S. Wagstaff, Jr. The new Mersenne conjecture. Amer. Math. Monthly, 96:125–128, 1980. [Bays and Hudson 2000a] C. Bays and R. Hudson. Zeroes of Dirichlet L-functions and irregularities in the distibution of primes. Math. Comp., 69:861–866, 2000. [Bays and Hudson 2000b] C. Bays and R. Hudson. A new bound for the smallest x with π(x) > li (x). Math. Comp., 69:1285–1296, 2000. [Bernstein 1997] D. Bernstein. Multidigit multiplication for mathematicians, 1997. em Advances Appl. Math., to appear. http://cr.yp.to/arith.html#m3. [Bernstein 1998] D. Bernstein. Bounding smooth integers (extended abstract). In [Buhler 1998], pages 128–130. [Bernstein 2003] D. Bernstein. Proving primality in essentially quartic time. http://cr.yp.to/ntheory.html#quartic. [Bernstein 2004a] D. Bernstein. Scaled remainder trees. http://cr.yp.to/papers.html#scaledmod.

REFERENCES 549<br />

[Bach and Shallit 1996] E. Bach and J. Shallit. Algorithmic Number Theory,<br />

volume I. MIT Press, 1996.<br />

[Baillie and Wagstaff 1980] R. Baillie and S. Wagstaff, Jr. Lucas pseudoprimes.<br />

Math. Comp., 35:1391–1417, 1980.<br />

[Bailey 1990] D. Bailey. FFTs in external or hierarchical memory. J. Supercomp.,<br />

4:23–35, 1990.<br />

[Bailey and Crandall 2001] D. Bailey and R. Crandall. On the random character of<br />

fundamental constant expansions, Experiment. Math., 10:175–190, 2001.<br />

[Bailey and Crandall 2002] D. Bailey and R. Crandall. Random generators and<br />

normal numbers. Experiment. Math., 11:527–546, 2002.<br />

[Bailey et al. 2003] D. Bailey, J. Borwein, R. Crandall, and C. Pomerance. On the<br />

binary expansions of algebraic numbers. Journal de Theorie des Nombres,<br />

Bordeau (to appear), 2003.<br />

[Balasubramanian and Nagaraj 1997] R. Balasubramanian and S. Nagaraj.<br />

Density of Carmichael numbers with three prime factors. Math. Comp.,<br />

66:1705–1708, 1997.<br />

[Balazard et al. 1999] M. Balazard, E. Saias, and M. Yor. Notes sur la fonction ζ<br />

de Riemann. II. Adv. Math., 143:284–287, 1999.<br />

[Balog 1989] A. Balog. On a variant of the Piatetski-Shapiro prime number<br />

theorem. In Groupe de travail en théorie analytique et élementaire des<br />

nombres, 1987–1988, volume 89-01 of Publ. Math. Orsay, pages 3–11.<br />

Univ. Paris XI, Orsay, 1989.<br />

[Barrett 1987] P. Barrett. Implementing the Rivest Shamir and Adleman public<br />

key encryption algorithm on a standard digital signal processor. In<br />

A. Odlyzko, editor, Advances in Cryptology, Proc. Crypto ’86, volume 263<br />

of Lecture Notes in Computer Science, pages 311–323. Springer–Verlag,<br />

1987.<br />

[Bateman et al. 1989] P. Bateman, J. Selfridge, and S. Wagstaff, Jr. The new<br />

Mersenne conjecture. Amer. Math. Monthly, 96:125–128, 1980.<br />

[Bays and Hudson 2000a] C. Bays and R. Hudson. Zeroes of Dirichlet L-functions<br />

and irregularities in the distibution of primes. Math. Comp., 69:861–866,<br />

2000.<br />

[Bays and Hudson 2000b] C. Bays and R. Hudson. A new bound for the smallest<br />

x with π(x) > li (x). Math. Comp., 69:1285–1296, 2000.<br />

[Bernstein 1997] D. Bernstein. Multidigit multiplication for mathematicians, 1997.<br />

em Advances Appl. Math., to appear. http://cr.yp.to/arith.html#m3.<br />

[Bernstein 1998] D. Bernstein. Bounding smooth integers (extended abstract). In<br />

[Buhler 1998], pages 128–130.<br />

[Bernstein 2003] D. Bernstein. Proving primality in essentially quartic time.<br />

http://cr.yp.to/ntheory.html#quartic.<br />

[Bernstein 2004a] D. Bernstein. Scaled remainder trees.<br />

http://cr.yp.to/papers.html#scaledmod.

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