## Experimental mathematics: Examples, methods and implications (2005)

### Cached

### Download Links

Citations: | 15 - 3 self |

### BibTeX

@MISC{Bailey05experimentalmathematics:,

author = {David H. Bailey and Jonathan M. Borwein},

title = {Experimental mathematics: Examples, methods and implications},

year = {2005}

}

### OpenURL

### Abstract

### Citations

2046 | Handbook of mathematical functions - Abramowitz, Stegun - 1972 |

763 |
Cramming more components onto integrated circuits
- Moore
- 1965
(Show Context)
Citation Context ...d to continue, if not to increase. Over the longer term, the rate of increase is a bit more uncertain, although there is no reason to believe it will not remain nearly constant for at least 10 years. =-=[29]-=- Nearly forty years later, we observe a record of sustained exponential progress that has no peer in the history of technology. Hardware progress alone has transformed mathematical computations that w... |

311 | Uniform Distribution of Sequences - Kupiers, Niederreiter - 1974 |

309 | An introduction to Numerical Analysis - Atkinson - 1989 |

101 | On the rapid computation of various polylogarithmic constants
- Bailey, Borwein, et al.
- 1997
(Show Context)
Citation Context ... 2 1 1 − − − . 8k + 4 8k + 5 8k + 6 This formula permits one to directly calculate binary or hexadecimal digits beginning at the n-th digit, without needing to calculate any of the first n − 1 digits =-=[8]-=-, using a simple scheme that requires very little memory and no multiple-precision arithmetic software. It is easiest to see how this individual digitcalculating scheme works by illustrating it for a ... |

65 | Analysis of PSLQ, an integer relation finding algorithm
- Ferguson, Bailey, et al.
- 1999
(Show Context)
Citation Context ... one recent integer relation detection computation, 50,000-digit arithmetic was required to obtain the result [9]. At the present time, the best-known integer relation algorithm is the PSLQ algorithm =-=[26]-=- of mathematician-sculptor Helaman Ferguson, who, together with his wife, Claire, received the 2002 Communications Award of the Joint Policy Board for Mathematics (AMS-MAA-SIAM). Simple formulations o... |

59 | Special values of multiple polylogarithms
- Borwein, Bradley, et al.
(Show Context)
Citation Context ... integer arguments in higher dimensions. The analytic evaluation of such sums has relied on fast methods for computing their numerical values. One scheme, based on Hölder Convolution, is discussed in =-=[22]-=- and implemented in EZFace+, an online tool available at http://www.cecm.sfu. ca/projects/ezface+. We will illustrate its application to one specific case, namely the analytic identification of the su... |

33 |
Average case error estimates for the strong probable prime test
- Landrock, Pomerance
(Show Context)
Citation Context ...93 that if an integer n has k bits, then the probability that it is prime, provided it passes the most commonly used probabilistic test, is greater than 1 − k242−√k , and for certain k is even higher =-=[25]-=-. For instance, if n has 500 bits, then this probability is greater than 1 − 1/428m . Thus a 500-bit integer that passes this test even once is prime with prohibitively safe odds: the chance of a fals... |

23 | Random generators and normal numbers - BAILEY, CRANDALL |

20 | A Comparison of Three High-Precision Quadrature Schemes
- Bailey, Li
(Show Context)
Citation Context ...grals, typically producing up to 1000 digit accuracy in just a few seconds’ (or at most a few minutes’) run time on a 2004-era personal computer, and that are also well suited for parallel processing =-=[13]-=-, [14], [16, p. 312]. These schemes are based on the Euler-Maclaurin summation formula [3, p. 180], which can be stated as follows: Let m ≥ 0 and n ≥ 1 be integers, and define h = (b − a)/n and xj = a... |

20 | Experimentation in Mathematics: Computational Paths to Discovery, A K Peters - Borwein, Bailey, et al. - 2004 |

16 |
Parallel Integer Relation Detection
- Bailey, Broadhurst
(Show Context)
Citation Context ...10]. The PSLQ algorithm, together with related lattice reduction schemes such as LLL, was recently named one of ten “algorithms of the century” by the publication Computing in Science and Engineering =-=[4]-=-. PSLQ or a variant is implemented in current releases of most computer algebra systems. Arbitrary Digit Calculation Formulas The best-known application of PSLQ in experimental mathematics is the 1995... |

16 | Mathematics by Experiment
- Borwein, Bailey
- 2004
(Show Context)
Citation Context ...η + 9b2 . η + ... This continued fraction arises in Ramanujan’s Notebooks. He discovered the beautiful fact that ( Rη (a, b) + Rη (b, a) a + b = Rη 2 2 , √ ) ab . The authors wished to record this in =-=[15]-=- and wished to computationally check the identity. A first attempt to numerically compute R1 (1, 1) directly failed miserably, and with some effort only three reliable digits were obtained: 0.693 ....... |

10 | A Seventeenth-Order Polylogarithm Ladder
- Bailey, Broadhurst
- 1999
(Show Context)
Citation Context ... reason for the interest in very high-precision arithmetic in experimental mathematics. In one recent integer relation detection computation, 50,000-digit arithmetic was required to obtain the result =-=[9]-=-. At the present time, the best-known integer relation algorithm is the PSLQ algorithm [26] of mathematician-sculptor Helaman Ferguson, who, together with his wife, Claire, received the 2002 Communica... |

9 |
Randomness and complexity in pure mathematics
- Chaitin
- 1994
(Show Context)
Citation Context ...n is whether these experimental methods may be capable of discovering facts that are fundamentally beyond the reach of formal proof methods, which, due to Gödel’s result, we know must exist; see also =-=[24]-=-. One interesting example, which has arisen in our work, is the following. We mentioned in Section 3 the fact that the question of the 2-normality of π reduces to the question of whether the chaotic i... |

8 |
Finding and Excluding b-ary Machin-Type BBP Formulae
- Borwein, Galway, et al.
(Show Context)
Citation Context ... other than powers of two (although this does not rule out some other scheme for calculating individual digits). We will sketch this result here. Full details and some related results can be found in =-=[20]-=-. In the following, ℜ(z) and ℑ(z) denote the real and imaginary parts of z, respectively. The integer b>1 is not a proper power if it cannot be written as c m for any integers c and m >1. We will use ... |

7 |
Definitely an integral
- Ahmed
- 2002
(Show Context)
Citation Context ...han proven knowledge. As one example, recently the present authors, together with Greg Fee of Simon Fraser University in Canada, were inspired by a recent problem in the American Mathematical Monthly =-=[2]-=-. They found by using a tanh-sinh quadrature program, together with a PSLQ integer relation detection program, that if C(a) is defined by then C(a) = ∫ 1 C(0) = π log 2/8 + G/2, 0 arctan( √ x2 + a2 ) ... |

6 |
On the Ramanujan AGM fraction. Part II: The complex-parameter case, Experiment
- BORWEIN, CRANDALL
(Show Context)
Citation Context ... 4 ) 1 , 4 ) = 1 2 ∑ n∈Z sech(nπ) 1 + n 2 is close to closed. Here β denotes the classical Beta function. It would be pleasant to find a direct proof of (23). Further details are to be found in [19], =-=[17]-=-, [16]. Study of these Ramanujan continued fractions has been facilitated by examining the closely related dynamical system t0 = 1,t1 = 1, and MAY 2005 NOTICES OF THE AMS 509There are some exceptiona... |

5 |
On the Ramanujan AGM fraction. Part I: The real-parameter case, Experiment
- BORWEIN, CRANDALL, et al.
(Show Context)
Citation Context ....html, that the quotient is π/2 − log(1 + √ 2). Thus we conclude, experimentally, that Figure 2. Dynamics and attractors of various iterations. R(2) = √ 2[π/2 − log(1 + √ 2)]. Indeed, it follows (see =-=[19]-=-) that ∫ 1 t R(a) = 2 1/a dt. 1 + t2 Note that R(1) = log 2. No nontrivial closed form is known for R(a, b) with a ≠ b, although R1 ( 1 4π β ( 1 4 ) √ 1 2 , , 4 8π β 0 ( 1 4 ) 1 , 4 ) = 1 2 ∑ n∈Z sech... |

3 | generators and normal numbers - Random |

3 | and Sinai Robins, Highly parallel, high-precision numerical quadrature, http://crd.lbl.gov/∼dhbailey/dhbpapers/quadparallel.pdf - Bailey - 2004 |

3 | Dynamics of generalizations of the AGM continued fraction of Ramanujan. Part I: Divergence, http://www.cs.dal.ca/~jborwein/BLuke.pdf
- BORWEIN, LUKE
- 2004
(Show Context)
Citation Context ...amanujan continued fractions) is now quite well understood. These studies have ventured into matrix theory, real analysis, and even the theory of martingales from probability theory [19], [17], [18], =-=[23]-=-. where θ := arg Γ ((1 + i)/2). Analysis is easy given the following striking hypergeometric parametrization of (24) when a = b ≠ 0 (see [18]), which was both experimentally discovered and is computer... |

2 |
On the dynamics of certain recurrence relations, Ramanujan
- BORWEIN, CRANDALL, et al.
- 2005
(Show Context)
Citation Context ...inued fractions has been facilitated by examining the closely related dynamical system t0 = 1,t1 = 1, and MAY 2005 NOTICES OF THE AMS 509There are some exceptional cases. JacobsenMasson theory [17], =-=[18]-=- shows that the even/odd fractions for R1(i,i) behave “chaotically”; neither converge. Indeed, when a = b = i, (tn(i,i)) exhibit a fourfold quasi-oscillation, as n runs through values mod 4. Plotted v... |

2 | On two fundamental identities for Euler sums - Borwein, Bradley - 2005 |

1 | Eric Weisstein, Ten problems of experimental mathematics, http://crd.lbl.gov/∼dhbailey/dhbpapers/tenproblems.pdf - Bailey, Borwein, et al. - 2004 |

1 | Irreducible compexity in pure mathematics - Chaitin - 2004 |

1 | There’s plenty of room at the bottom, http://engr.smu.edu/ee/smuphotonics/ Nano/FeynmanPlentyofRoom.pdf - FEYNMAN - 1959 |

1 |
Ten problems of experimental mathematics, http://crd.lbl.gov/~dhbailey/ dhbpapers/tenproblems.pdf
- BAILEY, BORWEIN, et al.
- 2004
(Show Context)
Citation Context ...+ y 2 + (z − w) 2 dw dxdy dz 0 0 0 0 + 1 ∫ 1 ∫ 1 ∫ 1 ∫ 1 5 0 0 0 0 = 4 17 √ 2 √ 7 + 2 − 3 − 75 75 25 75 π + 7 25 log ( 1 + √ ) 2 + 7 25 log √ 1 + (y − u) 2 + (z − w) 2 du dw dy dz ( 7 + 4 √ 3 ) . See =-=[7]-=- for details and some additional examples. It is not known whether similar closed forms exist for higher-dimensional cubes. 508NOTICES OF THE AMS VOLUME 52, NUMBER 5Ramanujan’s AGM Continued Fraction... |

1 |
Random generators and normal
- BAILEY, CRANDALL
(Show Context)
Citation Context ...is equidistributed in the unit interval. For π, the associated sequence is x0 = 0 and { 120n xn = 16xn−1 + 2 − 89n + 16 512n4 − 1024n3 + 712n2 } . − 206n + 21 Full details of this result are given in =-=[11]-=- [15, Section 3.8]. It is difficult to know at the present time whether this result will lead to a full-fledged proof of normality for, say, π or log 2. However, this approach MAY 2005 NOTICES OF THE ... |

1 |
Highly parallel, highprecision numerical quadrature, http://crd. lbl.gov/~dhbailey/dhbpapers/quadparallel.pdf
- BAILEY, ROBINS
- 2004
(Show Context)
Citation Context ... typically producing up to 1000 digit accuracy in just a few seconds’ (or at most a few minutes’) run time on a 2004-era personal computer, and that are also well suited for parallel processing [13], =-=[14]-=-, [16, p. 312]. These schemes are based on the Euler-Maclaurin summation formula [3, p. 180], which can be stated as follows: Let m ≥ 0 and n ≥ 1 be integers, and define h = (b − a)/n and xj = a + jh ... |

1 |
On two fundamental identities for Euler sums, http://www. cs.dal.ca/~jborwein/z21.pdf
- BORWEIN, BRADLEY
- 2005
(Show Context)
Citation Context ...re, it turns out that Euler considered some related summations. Perhaps it was just as well that Borwein was not aware of these earlier results—and indeed of a large, quite deep and varied literature =-=[21]-=-— because pursuit of this and similar questions had led to a line of research that continues to the present day. First define the multi-zeta constant ζ(s1,s2, ··· ,sk) := ∑ n1>n2>···>n k>0 k∏ j=1 n −|... |