Home
Search Site
Largest
Finding
How Many?
Mersenne
Glossary
Prime Curios!
email list
FAQ
Prime Lists
Titans
Submit primes

This is the Prime Pages'
interface to our BibTeX database. Rather than being an exhaustive database,
it just lists the references we cite on these pages. Please let me know of any errors you notice.References: [ Home  Author index  Key index  Search ] All items with keys beginning with the letter(s): b
 Bach85
 E. Bach, Analytic methods in the analysis and design of numbertheoretic algorithms, A.C.M. Distinguished Dissertations The MIT Press, 1985. Cambridge, MA, pp. xiii+48, ISBN 0262022192. MR 87i:11185
 Bach90
 E. Bach, "Explicit bounds for primality testing and related problems," Math. Comp., 55:191 (1990) 355380. MR 91m:11096
 Bach97
 E. Bach, "The complexity of numbertheoretic constants," Inform. Process. Lett., 62:3 (1997) 145152. MR 98g:11148
 Bailey90
 D. Bailey, "FFTs in external or hierarchical memory," Journal of Supercomputing, 4:1 (1990) 2335.
 Baillie1979
 R. Baillie, "New primes of the form k · 2^{n} + 1," Math. Comp., 33:148 (October 1979) 13331336. MR 80h:10009 (Abstract available)
 Balog1990
 Balog, Antal, The prime ktuplets conjecture on average. In "Analytic number theory (Allerton {P}ark, {IL}, 1989)," Progr. Math. Vol, 85, Birkh\"auser Boston, Boston, MA, 1990. pp. 4775, MR 1084173
 BB2000
 S. Battiato and W. Borho, "Breeding amicable numbers in abundance. II," Math. Comp., 70:235 (2001) 13291333. MR 2002b:11011 (Abstract available) [See [BH1986].]
 BBC1999
 J. M. Borwein, D. M. Bradley and R. E. Crandall, "Computational strategies for the Riemann zeta function," J. Comput. Appl. Math., 121:12 (2000) 247296. Numerical analysis in the 20th century, Vol. I, Approximation. MR 2001h:11110
 BBCGP88
 P. Beauchemin, G. Brassard, C. Crépeau, C. Goutier and C. Pomerance, "The generation of random numbers that are probably prime," J. Cryptology, 1 (1988) 5364. MR 89g:11126
 BBLR1998
 E. Barcucci, S. Brunetti, A. Del Lungo and F. Del Ristoro, "A combinatorial interpretation of the recurrence f_{n+1}=6f_{n}f_{n1}," Discrete Math., 190 (1998) 235240. MR 99f:05001
 BCEM1993
 J. Buhler, R. Crandall, R. Ernvall and T. Metsänkylä, "Irregular primes and cyclotomic invariants to four million," Math. Comp., 61:203 (1993) 151153. MR 93k:11014
 BCEMS2000
 J. Buhler, R. Crandall, R. Ernvall, T. Metsankyla and M. Shokrollahi, "Irregular primes and cyclotomic invariants to 12 million," J. Symbolic Comput., 31:12 (2001) 8996. MR 2001m:11220
 BCLK2009
 A. Brunner, C. Caldwell, C. Lownsdale and D. Krywaruczenko, Generalizing sierpi\'nski numbers to base b. In "New Aspects of Analytic Number Theory (Kyoto October 2008 proceedings)," T. Komatsu editor, RIMS, Kyoto, 2009. pp. 6979,
 BCP82
 J. P. Buhler, R. E. Crandall and M. A. Penk, "Primes of the form n! ± 1 and 2 · 3 · 5 ^{...} p ± 1," Math. Comp., 38:158 (1982) 639643. Corrigendum in Math. Comp. 40 (1983), 727. MR 83c:10006
 BCR91
 R. P. Brent, G. L. Cohen and H. J. J. te Riele, "Improved techniques for lower bounds for odd perfect numbers," Math. Comp., 57:196 (1991) 857868. MR 92c:11004
 BCS1992
 J. P. Buhler, R. E. Crandall and R. W. Sompolski, "Irregular primes to one million," Math. Comp., 59:200 (1992) 717722. MR 93a:11106
 BCW1982
 R. Baillie, G. Cormack and H. C. Williams, "Corrigenda: "The problem of Sierpi\'nski concerning k· 2^{n}+1" [Math. Comp. 37 (1981), no. 155, 229231]," Math. Comp., 39:159 (1982) 308. MR 83a:10006b
 BCW81
 Baillie, R., Cormack, G. and Williams, H.C., "The problem of Sierpinski concerning k · 2^{n} + 1," Math. Comp., 37:155 (1981) 229231. MR 83a:10006a [Corrigenda: [BCW1982]]
 BD1974
 T. Bhargava and P. Doyle, "On the existence of absolute primes," Math. Mag., 47 (1974) 233. MR 49:10630
 Beeger50
 N. G. W. H. Beeger, "On composite numbers n for which a^{n1} ≡ 1 (mod n) for every a prime to n," Scripta Math., 16 (1950) 133135. MR 12,159e
 Beeger51
 N. G. W. H. Beeger, "On even numbers m dividing 2^{m}  2," Amer. Math. Monthly, 58 (1951) 553555. MR 13,320d
 Beiler1964
 A. Beiler, Recreations in the theory of numbers, Dover Pub., 1964. New York, NY,
 Benford1938
 F. Benford, "The law of anomalous numbers," Proc. Amer. Math. Soc., 78 (1938) 551572. [See [Newcomb1881]. Data partially questioned in [DF1979, p. 363].]
 Bernstein1998
 D. Bernstein, "Multidigit multiplication for mathematicians," Advances in Applied Mathematics, (1998) to appear? Preprint available from http://cr.yp.to/papers.html. (Abstract available) (Annotation available)
 Bernstein1998b
 D. Berstein, "Detecting perfect powers in essentially linear time," Math. Comp., 67:223 (1998) 12531283. Available from http://cr.yp.to/papers.html. MR 98j:11121 (Abstract available)
 Bernstein2000
 D. Bernstein, "How to find small factors of integers," Math. Comp., Submitted August 2000. Available online at ftp://koobera.math.uic.edu/www/papers.html..
 Bernstein2001
 D. Bernstein, "Circuits for integer factorization: a proposal," (2001) Available from http://cr.yp.to/factorization.html. (Abstract available)
 Bernstein2003
 D. J. Bernstein, "Proving primality in essentially quartic random time," (2003) Draft available from http://cr.yp.to/papers.html.
Abstract:
This paper presents an algorithm that, given a prime n, finds and verifies a proof of the primality of n in random time (lg n)^{4+o(1)}.
 Berrizbeitia2003
 P. Berrizbeitia, "Sharpening "Primes is in P" for a large family of numbers," (2003) Available from http://arxiv.org/abs/math.NT/0211334. (Annotation available)
 BG59
 C. L. Baker and F. J. Gruenberger, The first six million prime numbers, Microcard Foundation, 1959. Madison, Wisconsin,
 BH1986
 Borho, W. and Hoffmann, H., "Breeding amicable numbers in abundance," Math. Comp., 46:173 (1986) 281293. MR 87c:11003
 BH1993
 E. Bach and L. Huelsbergen, "Statistical evidence for small generating sets," Math. Comp., 61:203 (1993) 6982. MR1195432
 BH1996
 R. C. Baker and G. Harman, The BrunTitchmarsh theorem on average. In "Proc. Conf. in Honor of Heini Halberstam (Allerton Park, IL, 1995)," Progr. Math. Vol, 138, Birkhäuser Boston, Boston, MA, 1996. pp. 39103, MR 97h:11096
 BH2000
 C. Bays and R. Hudson, "A new bound for the smallest x with π(x)>li(x)," Math. Comp., 69:231 (2000) 12851296 (electronic). MR1752093
 BH2011
 Buhler, J. P. and Harvey, D., "Irregular primes to 163 million," Math. Comp., 80:276 (2011) 24352444. (http://dx.doi.org/10.1090/S002557182011024610) MR 2813369
 BH62
 P. T. Bateman and R. A. Horn, "A heuristic asymptotic formula concerning the distribution of prime numbers," Math. Comp., 16 (1962) 363367. MR 26:6139
 BH65
 P. T. Bateman and R. A. Horn, Primes represented by irreducible polynomials in one variable. In "Proc. Symp. Pure Math.," Vol, VIII, Amer. Math. Soc., 1965. Providence, RI, pp. 119132, MR 31:1234
 BH77
 C. Bayes and R. Hudson, "The segmented sieve of Eratosthenes and primes in arithmetic progression," Nordisk Tidskr. Informationsbehandling (BIT), 17:2 (1977) 121127. MR 56:5405
 BH90
 W. Bosma and M. P. van der Hulst, Faster primality testing. In "Advances in CryptologyEUROCRYPT '89 Proceedings," J. J. Quisquater and J. Vandewalle editors, SpringerVerlag, 1990. pp. 652656,
 BH96
 R. C. Baker and G. Harman, "The difference between consecutive primes," Proc. Lond. Math. Soc., series 3, 72 (1996) 261280. MR 96k:11111 (Abstract available)
 BHV2002
 Bilu, Yu., Hanrot, G. and Voutier, P. M., "Existence of primitive divisors of Lucas and Lehmer numbers," J. Reine Angew. Math., 539 (2001) 75122. With an appendix by M. Mignotte. MR1863855 [From the review: "This remarkable paper answers completely a one century old problem, by proving that, for any integer n>30, the nth element of any Lucas or Lehmer sequence has a primitive divisor."]
 Biermann1964
 K. R. Biermann, "Thomas Clausen: Mathematiker and Astronom," J. Reine Angew. Math., 216 (1964) 159197. MR 29:2153
 Bleichenbacher1996
 D. Bleichenbacher, "Efficiency and security of cryptosystems based on number theory," Ph.D. thesis, ETH Zürich, (1996)
 BLS75
 J. Brillhart, D. H. Lehmer and J. L. Selfridge, "New primality criteria and factorizations of 2^{m} ± 1," Math. Comp., 29 (1975) 620647. MR 52:5546 [The article for the classical (n^{2} 1) primality tests. Table errata in [Brillhart1982]]
 BLSTW88
 J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman and S. S. Wagstaff, Jr., Factorizations of b^{n} ± 1, b=2,3,5,6,7,10,12 up to high powers, Amer. Math. Soc., Providence RI, 1988. pp. xcvi+236, ISBN 0821850784. MR 90d:11009 (Annotation available)
 BLZ94
 J. Buchmann, J. Loho and J. Zayer, An implementation of the general number field sieve, Advances in CryptologyCrypto '93 (Santa Barbara, CA, 1993) SpringerVerlag, 1994. New York, NY, pp. 159165, MR 95e:11132
 BM80
 Brent, R. P. and McMillan, E. M., "Some new algorithms for highprecision computation of Euler's constant," Math. Comp., 34:149 (1980) 305312. MR 82g:10002
 BMS1988
 J. Brillhart, P. Montgomery and R. Silverman, "Tables of Fibonacci and Lucas factorizations," Math. Comp., 50 (1988) 251260. MR 89h:11002
 BMS88
 J. Brillhart, P. L. Montgomery and R. D. Silverman, "Tables of Fibonacci and Lucas factorizations," Math. Comp., 50 (1988) 251260, S1S15. MR 89h:11002 [See also [DK99].]
 BNPSSZ90
 J. Brown, L. C. Noll, B. K. Parady, J. F. Smith, G. W. Smith and S. E. Zarantonello, "Letter to the editor," Amer. Math. Monthly, 97 (1990) 214. [The prime 391581 · 2^{216193}  1 is announced]
 BNW87
 J. Bogoshi, K. Naidoo and J. Webb, "The oldest mathematical artefact," Math. Gazette, 71:458 (1987) 294. MR 89a:01003
 Bond84
 D. J. Bond, Practical primality testing. In "Proc. Int'l. Conf. Secure Communication Systems," IEE, 1984. pp. 5053,
 Bone1999
 C. Bone, "The Mersenne telescope," Sky \& Telescope, (September 1999) 130133. [A variant of a design by Mersenne. This one was built with a 30inch primary mirror.]
 Boorstin1983
 D. Boorstin, The discoverers, Random House Inc., New York, NY, 1983. ISBN 0394402294.
 Borning72
 A. Borning, "Some results for k! ± 1 and 2 · 3 · 5 ^{...} p ± 1," Math. Comp., 26 (1972) 567570. MR 46:7133
 Bosma2006
 Bosma, Wieb, Some computational experiments in number theory. In "Discovering mathematics with Magma," Algorithms Comput. Math. Vol, 19, Springer, 2006. Berlin, pp. 130, MR2278921
 Bosma93
 W. Bosma, "Explicit primality criteria for h · 2^{k} ± 1," Math. Comp., 61:203 (1993) 97109, S7S9. MR 94c:11005
 Bowen1964
 Bowen, Robert, "Mathematical Notes: The sequence ka^{n} + 1 composite for all n," Amer. Math. Monthly, 71:2 (1964) 175176. MR1532528
 BP2001
 R. Bhattacharjee and P. Pandey, "Primality testing," IIT Kanpur, (2001)
 BPR96
 R. Brent, A. van der Poorten and H. J. J. te Riele, A comparative study of algorithms for computing continued fractions of algebraic numbers. In "Algorithmic Number Theory, Second International Symposium," SpringerVerlag, Berlin, 1996. pp. 3749, ANTSII in Talence France, May 1823, 1996. MR 98c:11144
 BR98
 A. Björn and H. Riesel, "Factors of generalized Fermat numbers," Math. Comp., 67 (1998) 441446. MR 98e:11008 (Abstract available)
 Brent1982
 Brent, Richard P., "Succinct proofs of primality for the factors of some Fermat numbers," Math. Comp., 38:157 (1982) 253255. (http://dx.doi.org/10.2307/2007482) MR 637304
 Brent73
 R. P. Brent, "The first occurrence of large gaps between successive primes," Math. Comp., 27 (1973) 959963. MR 0330021
 Brent74
 R. P. Brent, "The distribution of small gaps between succesive primes," Math. Comp., 28 (1974) 315324. MR 48:8356
 Brent75
 R. P. Brent, "Irregularities in the distribution of primes and twin primes," Math. Comp., 29 (1975) 4356. MR 51:5522
 Bressoud89
 D. M. Bressoud, Factorizations and primality testing, SpringerVerlag, 1989. New York, NY, ISBN 0387970401. MR 91e:11150 [QA161.F3B73]
 Brillhart1982
 Brillhart, J., "Table errata: "New primality criteria and factorizations of 2^{m}± 1" [Math. Comp. 29 (1975), 620647. MR 52 \#5546. by the author, D. H. Lehmer and J. L. Selfridge," Math. Comp., 39:160 (1982) 747. MR 83j:10010
 Brillhart1999
 J. Brillhart, "Note on Fibonacci primality testing," Fibonacci Quart., 36:3 (1998) 222228. MR1627388
 Brooke1960
 M. Brooke, "On the digital roots of perfect numbers," Math. Mag., 34:2 (1960) 100. MR 23:A840
 Bruce93
 J. W. Bruce, "A really trivial proof of the lucaslehmer test," Amer. Math. Monthly, 100 (1993) 370371. MR 94c:11006 [See also [Rosen88].]
 Bruckman94a
 P. S. Bruckman, "On the infinitude of Lucas pseudoprimes," Fibonacci Quart., 32 (1994) 153154. MR 95d:11011
 Bruckman94b
 P. S. Bruckman, "Lucas pseudoprimes are odd," Fibonacci Quart., 32 (1994) 155157. MR 95d:11012
 Brun19
 V. Brun, "La serie 1/5 + 1/7 + [etc.] où les denominateurs sont "nombres premiers jumeaux" est convergente ou finie," Bull. Sci. Math., 43 (1919) 100104,124128.
 BS96
 E. Bach and J. Shallit, Algorithmic number theory, Foundations of Computing Vol, I: Efficient Algorithms, The MIT Press, Cambridge, MA, 1996. pp. xvi+512, MR 97e:11157 (Annotation available)
 BSW89
 P. T. Bateman, J. L. Selfridge and Wagstaff, Jr., S. S., "The new Mersenne conjecture," Amer. Math. Monthly, 96 (1989) 125128. MR 90c:11009
 Burthe1996
 R. Burthe, Jr., "Further investigations with the strong probable prime test," Math. Comp., 65:213 (1996) 373381. MR 96d:11137 (Abstract available)
 Burton80
 D. M. Burton, Elementary number theory, Allyn and Bacon, 1980. MR 81c:10001a [Burton's texts are excellent introductions to elementary number theory and its history.]
 Burton91
 D. Burton, The history of mathematics, Wm. C. Brown, 1991. MR 94b:01002
 Burton97
 D. M. Burton, Elementary number theory, Third edition, McGrawHill, 1997. [Burton's texts are excellent introductions to elementary number theory and its history.]
 BW2000
 D. Bressoud and S. Wagon, A course in computational number theory, Key College Publishing, 2000. MR 2001f:11200 (Annotation available)
 BW80
 R. Baillie and Wagstaff, Jr., S. S., "Lucas pseudoprimes," Math. Comp., 35 (1980) 13911417. MR 81j:10005
 BY88
 D. A. Buell and J. Young, "Some large primes and the Sierpinski problem," SRC techn. Rep. 88004, SuperComputing Res. Center, Lanham, MD, (1988)
