Results 1  10
of
12
Normal Numbers and Pseudorandom Generators
, 2011
"... For an integer b ≥ 2 a real number α is bnormal if, for all m> 0, every mlong string of digits in the baseb expansion of α appears, in the limit, with frequency b −m. Although almost all reals in [0, 1] are bnormal for every b, it has been rather difficult to exhibit explicit examples. No result ..."
Abstract

Cited by 7 (4 self)
 Add to MetaCart
For an integer b ≥ 2 a real number α is bnormal if, for all m> 0, every mlong string of digits in the baseb expansion of α appears, in the limit, with frequency b −m. Although almost all reals in [0, 1] are bnormal for every b, it has been rather difficult to exhibit explicit examples. No results whatsoever are known, one way or the other, for the class of “natural ” mathematical constants, such as π, e, √ 2 and log 2. In this paper, we summarize some previous normality results for a certain class of explicit reals, and then show that a specific member of this class, while provably 2normal, is provably not 6normal. We then show that a practical and reasonably effective pseudorandom number generator can be defined based on the binary digits of this constant, and conclude by sketching out some directions for further research.
Experimental Mathematics and Computational Statistics
, 2009
"... The field of statistics has long been noted for techniques to detect patterns and regularities in numerical data. In this article we explore connections between statistics and the emerging field of “experimental mathematics.” These includes both applications of experimental mathematics in statistics ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
The field of statistics has long been noted for techniques to detect patterns and regularities in numerical data. In this article we explore connections between statistics and the emerging field of “experimental mathematics.” These includes both applications of experimental mathematics in statistics, as well as statistical methods applied to computational mathematics.
Nonnormality of Stoneham constants
, 2012
"... Previous studies have established that Stoneham’s constant α2,3 n≥1 1/(3n23n) is 2normal, or, in other words, every mlong string of binary digits appears in the binary expansion of α2,3 with precisely the expected limiting frequency 1/2m. A more recent finding is that this constant is provably not ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
Previous studies have established that Stoneham’s constant α2,3 n≥1 1/(3n23n) is 2normal, or, in other words, every mlong string of binary digits appears in the binary expansion of α2,3 with precisely the expected limiting frequency 1/2m. A more recent finding is that this constant is provably not 6normal. In this note we address the more general class of Stoneham constants αb,c = ∑ n≥1 1/(cnbcn), for coprime integers b ≥ 2 and c ≥ 2. It has been proven that αb,c is bnormal, but not bcnormal. Here we extend this finding by showing that αb,c is not Bnormal, where B = bpcqr, for integers b and c as above, p, q, r ≥ 1, neither b nor c divide r, and the condition D = cq/pr1/p /bc−1 < 1 is satisfied. It is not known whether or not this is a complete catalog of bases to which αb,c is nonnormal. We also show that the sum of two Bnonnormal Stoneham constants as defined above is Bnonnormal.
The Computation of Previously Inaccessible Digits of π 2 and Catalan’s Constant
, 2011
"... The admirable number pi: three point one four one. All the following digits are also initial, ..."
Abstract
 Add to MetaCart
The admirable number pi: three point one four one. All the following digits are also initial,
Normality and the Digits of π
"... The question of whether (and why) the digits of wellknown constants of mathematics are statistically random in some sense has long fascinated mathematicians. Indeed, one prime motivation in computing and analyzing digits of π is to explore the ageold ..."
Abstract
 Add to MetaCart
The question of whether (and why) the digits of wellknown constants of mathematics are statistically random in some sense has long fascinated mathematicians. Indeed, one prime motivation in computing and analyzing digits of π is to explore the ageold
Walking on real numbers
, 2012
"... Motivated by the desire to visualize large mathematical data sets, especially in number theory, we offer various tools for representing floating point numbers as planar (or three dimensional) walks and for quantitatively measuring their “randomness.” 1 ..."
Abstract
 Add to MetaCart
Motivated by the desire to visualize large mathematical data sets, especially in number theory, we offer various tools for representing floating point numbers as planar (or three dimensional) walks and for quantitatively measuring their “randomness.” 1
Nonnormality of Stoneham constants
, 2012
"... This paper examines “Stoneham constants, ” namely real numbers of the form αb,c = ∑ n≥1 1/(cnbcn), for coprime integers b ≥ 2 and c ≥ 2. These are of interest because, according to previous studies, αb,c is known to be bnormal, meaning that every mlong string of baseb digits appears in the baseb ..."
Abstract
 Add to MetaCart
This paper examines “Stoneham constants, ” namely real numbers of the form αb,c = ∑ n≥1 1/(cnbcn), for coprime integers b ≥ 2 and c ≥ 2. These are of interest because, according to previous studies, αb,c is known to be bnormal, meaning that every mlong string of baseb digits appears in the baseb expansion of the constant with precisely the limiting frequency b−m. So, for example, the constant α2,3 = ∑ n≥1 1/(3n23n) is 2normal. More recently it was established that αb,c is not bcnormal, so, for example, α2,3 is provably not 6normal. In this paper, we extend these findings by showing that αb,c is not Bnormal, where B = bpcqr, for integers b and c as above, p, q, r ≥ 1, neither b nor c divide r, and the condition D = cq/pr1/p /bc−1 < 1 is satisfied. It is not known whether or not this is a complete catalog of bases to which αb,c is nonnormal. We also show that the sum of two Bnonnormal Stoneham constants as defined above, subject to some restrictions, is Bnonnormal. 1
Pi Day is upon us again and we still do not know if Pi is normal
, 2013
"... The digits of π have intrigued both the public and research mathematicians from the beginning of time. This article briefly reviews the history of this venerable constant, and then describes some recent research on the question of whether π is normal, or, in other words, whether its digits are stati ..."
Abstract
 Add to MetaCart
The digits of π have intrigued both the public and research mathematicians from the beginning of time. This article briefly reviews the history of this venerable constant, and then describes some recent research on the question of whether π is normal, or, in other words, whether its digits are statistically random in a specific sense. 1 Pi and its day in modern popular culture The number π, unique among the pantheon of mathematical constants, captures the fascination both of the public and of professional mathematicians. Algebraic constants such as √ 2 are easier to explain and to calculate to high accuracy (e.g., using a simple Newton iteration scheme). The constant e is pervasive in physics and chemistry, and even appears in financial mathematics. Logarithms are ubiquitous in the social sciences. But none of these other constants has ever gained much traction in the popular culture. In contrast, we see π at every turn. In an early scene of Ang Lee’s 2012 movie adaptation of Yann Martel’s awardwinning book The Life of Pi, the title character Piscine (“Pi”) Molitor writes hundreds of digits of the decimal expansion of π on a blackboard to impress his teachers and schoolmates, who chant along with every digit. 1 This has even led to humorous takeoffs such as a 2013 Scott Hilburn cartoon entitled “Wife of Pi, ” which depicts a 4 figure seated next to a π figure, telling their marriage counselor “He’s irrational and he goes on and on. ” [21].
AN EXPERIMENTAL INVESTIGATION OF THE NORMALITY OF IRRATIONAL ALGEBRAIC NUMBERS
"... Abstract. We investigate the distribution of digits of large prefixes of the expansion of irrational algebraic numbers to different bases. We compute 2·3 18 bits of the binary expansions (corresponding to 2.33·10 8 decimals) of the 39 least PisotVijayaraghavan numbers, the 47 least known Salem numb ..."
Abstract
 Add to MetaCart
Abstract. We investigate the distribution of digits of large prefixes of the expansion of irrational algebraic numbers to different bases. We compute 2·3 18 bits of the binary expansions (corresponding to 2.33·10 8 decimals) of the 39 least PisotVijayaraghavan numbers, the 47 least known Salem numbers, the least 20 square roots of positive integers that are not perfect squares, and 15 randomly generated algebraic irrationals. We employ these to compute the generalized serial statistics (roughly, the variant of the χ 2statistic apt for distribution of sequences of characters) of the distributions of digit blocks for each number to bases 2, 3, 5, 7 and 10, as well as the maximum relative frequency deviation from perfect equidistribution. We use the two statistics to perform tests at significance level α =0.05, respectively, maximum deviation threshold α =0.05. Our results suggest that if Borel’s conjecture—that all irrational algebraic numbers are normal—is true, then it may have an empirical base: The distribution of digits in algebraic numbers appears close to equidistribution for large
Tools for visualizing real numbers: Planar number walks
, 2012
"... Motivated by the desire to visualize large mathematical data sets, especially in number theory, we offer various tools for representing floating point numbers as planar (or three dimensional) walks and for quantitatively measuring their “randomness.” 1 ..."
Abstract
 Add to MetaCart
Motivated by the desire to visualize large mathematical data sets, especially in number theory, we offer various tools for representing floating point numbers as planar (or three dimensional) walks and for quantitatively measuring their “randomness.” 1