Reference Database
(references for the Prime Pages)
The Prime Pages

Search Site

How Many?


Prime Curios!
e-mail list

Prime Lists

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

E. Bach, Analytic methods in the analysis and design of number-theoretic algorithms, A.C.M. Distinguished Dissertations The MIT Press, 1985.  Cambridge, MA, pp. xiii+48, ISBN 0-262-02219-2. MR 87i:11185
E. Bach, "Explicit bounds for primality testing and related problems," Math. Comp., 55:191 (1990) 355--380.  MR 91m:11096
E. Bach, "The complexity of number-theoretic constants," Inform. Process. Lett., 62:3 (1997) 145--152.  MR 98g:11148
D. Bailey, "FFTs in external or hierarchical memory," Journal of Supercomputing, 4:1 (1990) 23--35.
R. Baillie, "New primes of the form k · 2n + 1," Math. Comp., 33:148 (October 1979) 1333--1336.  MR 80h:10009 (Abstract available)
Balog, Antal, The prime k-tuplets conjecture on average.  In "Analytic number theory (Allerton {P}ark, {IL}, 1989)," Progr. Math. Vol, 85, Birkh\"auser Boston, 1990.  Boston, MA, pp. 47--75, MR 1084173
S. Battiato and W. Borho, "Breeding amicable numbers in abundance. II," Math. Comp., 70:235 (2001) 1329--1333.  MR 2002b:11011 (Abstract available) [See [BH1986].]
J. M. Borwein, D. M. Bradley and R. E. Crandall, "Computational strategies for the Riemann zeta function," J. Comput. Appl. Math., 121:1--2 (2000) 247--296.  Numerical analysis in the 20th century, Vol. I, Approximation.  MR 2001h:11110
P. Beauchemin, G. Brassard, C. Crépeau, C. Goutier and C. Pomerance, "The generation of random numbers that are probably prime," J. Cryptology, 1 (1988) 53--64.  MR 89g:11126
E. Barcucci, S. Brunetti, A. Del Lungo and F. Del Ristoro, "A combinatorial interpretation of the recurrence fn+1=6fn-fn-1," Discrete Math., 190 (1998) 235--240.  MR 99f:05001
J. Buhler, R. Crandall, R. Ernvall and T. Metsänkylä, "Irregular primes and cyclotomic invariants to four million," Math. Comp., 61:203 (1993) 151--153.  MR 93k:11014
J. Buhler, R. Crandall, R. Ernvall, T. Metsankyla and M. Shokrollahi, "Irregular primes and cyclotomic invariants to 12 million," J. Symbolic Comput., 31:1--2 (2001) 89--96.  MR 2001m:11220
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. 69--79,
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) 639--643.  Corrigendum in Math. Comp. 40 (1983), 727.  MR 83c:10006
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) 857--868.  MR 92c:11004
J. P. Buhler, R. E. Crandall and R. W. Sompolski, "Irregular primes to one million," Math. Comp., 59:200 (1992) 717--722.  MR 93a:11106
R. Baillie, G. Cormack and H. C. Williams, "Corrigenda: "The problem of Sierpi\'nski concerning k· 2n+1" [Math. Comp. 37 (1981), no. 155, 229--231]," Math. Comp., 39:159 (1982) 308.  MR 83a:10006b
Baillie, R., Cormack, G. and Williams, H.C., "The problem of Sierpinski concerning k · 2n + 1," Math. Comp., 37:155 (1981) 229--231.  MR 83a:10006a [Corrigenda: [BCW1982]]
T. Bhargava and P. Doyle, "On the existence of absolute primes," Math. Mag., 47 (1974) 233.  MR 49:10630
N. G. W. H. Beeger, "On composite numbers n for which an-1 ≡ 1 (mod n) for every a prime to n," Scripta Math., 16 (1950) 133--135.  MR 12,159e
N. G. W. H. Beeger, "On even numbers m dividing 2m - 2," Amer. Math. Monthly, 58 (1951) 553--555.  MR 13,320d
A. Beiler, Recreations in the theory of numbers, Dover Pub., 1964.  New York, NY,
F. Benford, "The law of anomalous numbers," Proc. Amer. Math. Soc., 78 (1938) 551--572. [See [Newcomb1881]. Data partially questioned in [DF1979, p. 363].]
D. Bernstein, "Multidigit multiplication for mathematicians," Advances in Applied Mathematics, (1998) to appear? Preprint available from (Abstract available) (Annotation available)
D. Berstein, "Detecting perfect powers in essentially linear time," Math. Comp., 67:223 (1998) 1253--1283.  Available from 98j:11121 (Abstract available)
D. Bernstein, "How to find small factors of integers," Math. Comp., Submitted August 2000. Available on-line at
D. Bernstein, "Circuits for integer factorization: a proposal," (2001) Available from (Abstract available)
D. J. Bernstein, "Proving primality in essentially quartic random time," (2003) Draft available from
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).
P. Berrizbeitia, "Sharpening "Primes is in P" for a large family of numbers," (2003) Available from (Annotation available)
C. L. Baker and F. J. Gruenberger, The first six million prime numbers, Microcard Foundation, Madison, Wisconsin, 1959.
Borho, W. and Hoffmann, H., "Breeding amicable numbers in abundance," Math. Comp., 46:173 (1986) 281--293.  MR 87c:11003
E. Bach and L. Huelsbergen, "Statistical evidence for small generating sets," Math. Comp., 61:203 (1993) 69--82.  MR1195432
R. C. Baker and G. Harman, The Brun-Titchmarsh 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. 39--103, MR 97h:11096
C. Bays and R. Hudson, "A new bound for the smallest x with π(x)>li(x)," Math. Comp., 69:231 (2000) 1285--1296 (electronic).  MR1752093
Buhler, J. P. and Harvey, D., "Irregular primes to 163 million," Math. Comp., 80:276 (2011) 2435--2444.  ( MR 2813369
P. T. Bateman and R. A. Horn, "A heuristic asymptotic formula concerning the distribution of prime numbers," Math. Comp., 16 (1962) 363-367.  MR 26:6139
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. 119-132, MR 31:1234
C. Bayes and R. Hudson, "The segmented sieve of Eratosthenes and primes in arithmetic progression," Nordisk Tidskr. Informationsbehandling (BIT), 17:2 (1977) 121--127.  MR 56:5405
W. Bosma and M. P. van der Hulst, Faster primality testing.  In "Advances in Cryptology--EUROCRYPT '89 Proceedings," J. J. Quisquater and J. Vandewalle editors, Springer-Verlag, 1990.  pp. 652--656,
R. C. Baker and G. Harman, "The difference between consecutive primes," Proc. Lond. Math. Soc., series 3, 72 (1996) 261--280.  MR 96k:11111 (Abstract available)
Bilu, Yu., Hanrot, G. and Voutier, P. M., "Existence of primitive divisors of Lucas and Lehmer numbers," J. Reine Angew. Math., 539 (2001) 75--122.  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 n-th element of any Lucas or Lehmer sequence has a primitive divisor."]
K. R. Biermann, "Thomas Clausen: Mathematiker and Astronom," J. Reine Angew. Math., 216 (1964) 159--197.  MR 29:2153
D. Bleichenbacher, "Efficiency and security of cryptosystems based on number theory," Ph.D. thesis, ETH Zürich, (1996)
J. Brillhart, D. H. Lehmer and J. L. Selfridge, "New primality criteria and factorizations of 2m ± 1," Math. Comp., 29 (1975) 620--647.  MR 52:5546 [The article for the classical (n2 -1) primality tests. Table errata in [Brillhart1982]]
J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman and S. S. Wagstaff, Jr., Factorizations of bn ± 1, b=2,3,5,6,7,10,12 up to high powers, Amer. Math. Soc., Providence RI, 1988.  pp. xcvi+236, ISBN 0-8218-5078-4. MR 90d:11009 (Annotation available)
J. Buchmann, J. Loho and J. Zayer, An implementation of the general number field sieve, Advances in Cryptology--Crypto '93 (Santa Barbara, CA, 1993) Springer-Verlag, New York, NY, 1994.  pp. 159--165, MR 95e:11132
Brent, R. P. and McMillan, E. M., "Some new algorithms for high-precision computation of Euler's constant," Math. Comp., 34:149 (1980) 305--312.  MR 82g:10002
J. Brillhart, P. Montgomery and R. Silverman, "Tables of Fibonacci and Lucas factorizations," Math. Comp., 50 (1988) 251--260.  MR 89h:11002
J. Brillhart, P. L. Montgomery and R. D. Silverman, "Tables of Fibonacci and Lucas factorizations," Math. Comp., 50 (1988) 251--260, S1--S15.  MR 89h:11002 [See also [DK99].]
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 · 2216193 - 1 is announced]
J. Bogoshi, K. Naidoo and J. Webb, "The oldest mathematical artefact," Math. Gazette, 71:458 (1987) 294.  MR 89a:01003
D. J. Bond, Practical primality testing.  In "Proc. Int'l. Conf. Secure Communication Systems," IEE, 1984.  pp. 50--53,
C. Bone, "The Mersenne telescope," Sky \& Telescope, (September 1999) 130--133. [A variant of a design by Mersenne. This one was built with a 30-inch primary mirror.]
D. Boorstin, The discoverers, Random House Inc., 1983.  New York, NY, ISBN 0-394-40229-4.
A. Borning, "Some results for k! ± 1 and 2 · 3 · 5 ... p ± 1," Math. Comp., 26 (1972) 567--570.  MR 46:7133
Bosma, Wieb, Some computational experiments in number theory.  In "Discovering mathematics with Magma," Algorithms Comput. Math. Vol, 19, Springer, Berlin, 2006.  pp. 1--30, MR2278921
W. Bosma, "Explicit primality criteria for h · 2k ± 1," Math. Comp., 61:203 (1993) 97--109, S7--S9.  MR 94c:11005
Bowen, Robert, "Mathematical Notes: The sequence kan + 1 composite for all n," Amer. Math. Monthly, 71:2 (1964) 175--176.  MR1532528
R. Bhattacharjee and P. Pandey, "Primality testing," IIT Kanpur, (2001)
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," Springer-Verlag, Berlin, 1996.  pp. 37--49, ANTS-II in Talence France, May 18-23, 1996.  MR 98c:11144
A. Björn and H. Riesel, "Factors of generalized Fermat numbers," Math. Comp., 67 (1998) 441--446.  MR 98e:11008 (Abstract available)
Brent, Richard P., "Succinct proofs of primality for the factors of some Fermat numbers," Math. Comp., 38:157 (1982) 253--255.  ( MR 637304
R. P. Brent, "The first occurrence of large gaps between successive primes," Math. Comp., 27 (1973) 959-963.  MR 0330021
R. P. Brent, "The distribution of small gaps between succesive primes," Math. Comp., 28 (1974) 315--324.  MR 48:8356
R. P. Brent, "Irregularities in the distribution of primes and twin primes," Math. Comp., 29 (1975) 43--56.  MR 51:5522
D. M. Bressoud, Factorizations and primality testing, Springer-Verlag, 1989.  New York, NY, ISBN 0387970401. MR 91e:11150 [QA161.F3B73]
Brillhart, J., "Table errata: "New primality criteria and factorizations of 2m± 1" [Math. Comp. 29 (1975), 620--647. MR 52 \#5546. by the author, D. H. Lehmer and J. L. Selfridge," Math. Comp., 39:160 (1982) 747.  MR 83j:10010
J. Brillhart, "Note on Fibonacci primality testing," Fibonacci Quart., 36:3 (1998) 222--228.  MR1627388
M. Brooke, "On the digital roots of perfect numbers," Math. Mag., 34:2 (1960) 100.  MR 23:A840
J. W. Bruce, "A really trivial proof of the lucas-lehmer test," Amer. Math. Monthly, 100 (1993) 370-371.  MR 94c:11006 [See also [Rosen88].]
P. S. Bruckman, "On the infinitude of Lucas pseudoprimes," Fibonacci Quart., 32 (1994) 153--154.  MR 95d:11011
P. S. Bruckman, "Lucas pseudoprimes are odd," Fibonacci Quart., 32 (1994) 155--157.  MR 95d:11012
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) 100--104,124--128.
E. Bach and J. Shallit, Algorithmic number theory, Foundations of Computing Vol, I: Efficient Algorithms, The MIT Press, 1996.  Cambridge, MA, pp. xvi+512, MR 97e:11157 (Annotation available)
P. T. Bateman, J. L. Selfridge and Wagstaff, Jr., S. S., "The new Mersenne conjecture," Amer. Math. Monthly, 96 (1989) 125-128.  MR 90c:11009
R. Burthe, Jr., "Further investigations with the strong probable prime test," Math. Comp., 65:213 (1996) 373--381.  MR 96d:11137 (Abstract available)
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.]
D. Burton, The history of mathematics, Wm. C. Brown, 1991.  MR 94b:01002
D. M. Burton, Elementary number theory, Third edition, McGraw-Hill, 1997. [Burton's texts are excellent introductions to elementary number theory and its history.]
D. Bressoud and S. Wagon, A course in computational number theory, Key College Publishing, 2000.  MR 2001f:11200 (Annotation available)
R. Baillie and Wagstaff, Jr., S. S., "Lucas pseudoprimes," Math. Comp., 35 (1980) 1391-1417.  MR 81j:10005
D. A. Buell and J. Young, "Some large primes and the Sierpinski problem," SRC techn. Rep. 88004, Super-Computing Res. Center, Lanham, MD, (1988)
Prime Pages' Home
Another prime page by Chris K. Caldwell