Normality Analysis of Current World Record Computations for Catalan's Constant and Arc Length of a Lemniscate with a=1
NN ORMALITY A NALYSIS OF C URRENT W ORLD R ECORD C OMPUTATIONS FOR C ATALAN ’ S C ONSTANT AND A RC L ENGTHOF A L EMNISCATE WITH a = 1 Seungmin Kim ∗ Korea Science Academy of KAIST, 105-47, Baegyanggwanmun-ro, Busanjin-gu, Busan, 47162, Republic of Korea [email protected]
September 8, 2019 A BSTRACT
Catalan’s constant and the lemniscate constants have been important mathematical constants ofinterest to the mathematical society, yet various properties are unknown. An important property ofsignificant mathematical constants is whether they are normal numbers. This paper evaluates thenormality of decimal and hexadecimal representations of current world record computations of digitsfor the Catalan’s constant (600,000,000,100 decimal digits and 498,289,214,317 hexadecimal digits)and the arc length of a lemniscate with a = 1 (600,000,000,000 decimal digits and 498,289,214,234hexadecimal digits). All analyzed frequencies have been found to be persistent to the conjecture ofCatalan’s constant and the arc length of a lemniscate with a = 1 being a normal number in bases 10and 16. K eywords Catalan’s constant · Lemniscate constant · Normal number · Digit analysis
MSC 11 · MSC 62 · MSC 68
A normal number at a base of b ∈ Z > is a real number such that each of the b digits is distributed in the equal naturaldensity of b . Similarly, for a length n ∈ Z > , all b n digit combinations in base b have the same natural density of b − n ,thus being equally likely in occurrence. This implies that no digit or combination of digits occur frequently than others.Very little computable mathematical constants have been proved as normal, although the mathematical society widelyassumes that computable mathematical constants such as π , e , and √ are normal.Catalan’s constant and the lemniscate constants have been crucial in the development of mathematics by themselvesand their computation methods, and further related discoveries implicate even greater insight. Catalan’s constant G isdefined as G = β (2) = (cid:80) ∞ n =0 ( − n (2 n +1) , and whether it is transcendental or even irrational remains an open questionin mathematics [Nesterenko, 2016]. The arc length of a lemniscate with a = 1 is defined as s = √ π [Γ( )] . Thelemniscate constant is defined as L = s = (cid:82) dt √ − t and has been proven as a transcendental number [Finch, 2003;Todd, 2000]. The values L = L and L = G , where G is Gauss’s constant, are sometimes respectively referred to asthe first and second lemniscate constants, both values also proven as transcendental [Finch, 2003; Todd, 2000].Catalan’s constant, the arc length of a lemniscate with a = 1 , and all lemniscate constants are unknown if it is a normalnumber. Currently, one of the only plausible methods to approximately verify sequences or irrational numbers arenormal is to perform statistical analysis on a large sample of the sequence or number. Because of this, new worldrecords of computed mathematical constants are used for checking statistical consistency for the normality of manyimportant mathematical constants. Trueb [Trueb, 2016] has performed an analysis of digit combinations from length ∗ Corresponding author, ORCID: 0000-0002-5461-330X a r X i v : . [ m a t h . G M ] S e p im, S.one to three on the first (cid:98) π e (cid:99) trillion digits of π and has verified that the variance of the frequencies overall complieswith the expected variance. New world record calculations for the Catalan’s constant (600,000,000,100 decimal digitsand 498,289,214,317 hexadecimal digits) and the arc length of a lemniscate with a = 1 (600,000,000,000 decimaldigits and 498,289,214,234 hexadecimal digits) as of July 2019 have recently been calculated and verified using the y-cruncher program [Yee, 2019; Kim, 2019a,b]. This study attempts to expand the insight of these two importantmathematical constants on the conjecture of the Catalan’s constant and the arc length of a lemniscate with a = 1 bystatistically analyzing the calculated world record digit computation results [Kim, 2019a,b]. Analysis of digits was done using a custom-coded Python 3 script, speed optimized using the Portable PyPy3.6 v7.1.1JIT compiler [squeaky-pl/portable-pypy, 2019] on CentOS 7.4 using an Intel Xeon (Skylake Purley) CPU. The Pythonscript was coded so that the integrity of arbitrary length digit combinations are ensured if single digit counts are correct,and the single-digit counts were initially cross-checked for verification using the Digit Viewer application of y-cruncher [Yee, 2019]. Statistical analysis and visualization were done using a modified version of the method and code used inTrueb [Trueb, 2016] using the CERN ROOT Toolkit [Brun and Rademakers, 1997] Release 5.34/38, compiled usingGCC version 7.4.0 on Ubuntu 18.04 LTS using a KVM virtual machine with an Intel Xeon (Haswell) CPU. Sourcecode used for data analysis was based on the ANSI C99 standard of the C language.Digit occurrences from length one to three were counted starting from the decimal point. The decimal and hexadecimalexpressions of the Catalan’s constant and the arc length of a lemniscate with a = 1 were respectively used for theanalysis of digit occurrences. The computation results for Catalan’s constant had 600,000,000,100 decimal digits and498,289,214,317 hexadecimal digits for each base expression after the decimal point, and the computation results forthe arc length of a lemniscate with a = 1 had 600,000,000,000 decimal digits and 498,289,214,234 hexadecimal digitsfor each base expression after the decimal point. The expected variance of the frequencies and the expected error ofthe variance has been calculated by assuming the frequencies followed a binomial distribution around the limitingfrequency of b − k [Trueb, 2016]. Accessory regions were plotted with the area between the two vertical lines closestto the center of the figure representing the region within one standard deviation and the remaining area between tworemaining vertical lines representing the region between one and two standard deviations [Trueb, 2016]. Figures 1 to 6 depict the frequency distributions of all sequences from a length of one to three for the decimal andhexadecimal representations of Catalan’s constant in the form of a histogram. Table 1 lists the predicted and actualvariances for the frequency distributions of Catalan’s constant.Figures 7 to 12 depict the frequency distributions of all sequences from a length of one to three for the decimal andhexadecimal representations of the arc length of a lemniscate with a = 1 in the form of a histogram. Table 2 lists thepredicted and actual variances for the frequency distributions of the arc length of a lemniscate with a = 1 . Entries 10Mean 01 − − Frequency0.099999 0.1 0.100001 N u m be r o f S equen c e s Entries 10Mean 01 − − Figure 1:
Frequencies of all combinations of length1 (digits 0–9) in the decimal representation of theCatalan’s constant.
Entries 100Mean 02 − − Frequency9.9995 10 10.0005 − × N u m be r o f S equen c e s Entries 100Mean 02 − − Figure 2:
Frequencies of all combinations of length 2(00–99) in the decimal representation of the Catalan’sconstant.2im, S.
Entries 1000Mean 03 − − Frequency0.9999 1 1.0001 − × N u m be r o f S equen c e s Entries 1000Mean 03 − − Figure 3:
Frequencies of all combinations of length3 (000–999) in the decimal representation of the Cata-lan’s constant.
Entries 16Mean 02 − − Frequency62.499 62.5 62.501 − × N u m be r o f S equen c e s Entries 16Mean 02 − − Figure 4:
Frequencies of all combinations of length 1(digits 0–F) in the hexadecimal representation of theCatalan’s constant.
Entries 256Mean 03 − − Frequency3.906 3.9062 3.9064 3.9066 − × N u m be r o f S equen c e s Entries 256Mean 03 − − Figure 5:
Frequencies of all combinations of length2 (00–FF) in the hexadecimal representation of theCatalan’s constant.
Entries 4096Mean 04 − − Frequency0.2441 0.24415 0.2442 − × N u m be r o f S equen c e s Entries 4096Mean 04 − − Figure 6:
Frequencies of all combinations of length3 (000–FFF) in the hexadecimal representation of theCatalan’s constant.
Table 1:
Predicted and actual variances of frequencies of all sequences of length 1–3 in the decimal and hexadecimalrepresentations of Catalan’s constant.Base Length of Sequence Predicted Variance and Error ofFrequencies Actual Variance of Frequencies Deviation [ σ ]10 1 (1 . ± . × − . × − .
10 2 (1 . ± . × − . × − − .
10 3 (1 . ± . × − . × − .
16 1 (1 . ± . × − . × − − .
16 2 (7 . ± . × − . × − − .
16 3 (4 . ± . × − . × − − . Entries 10Mean 01 − − Frequency0.099999 0.1 0.100001 N u m be r o f S equen c e s Entries 10Mean 01 − − Figure 7:
Frequencies of all combinations of length1 (digits 0–9) in the decimal representation of the arclength of a lemniscate with a = 1 . Entries 100Mean 02 − − Frequency9.9995 10 10.0005 − × N u m be r o f S equen c e s Entries 100Mean 02 − − Figure 8:
Frequencies of all combinations of length 2(00–99) in the decimal representation of the arc lengthof a lemniscate with a = 1 . Entries 1000Mean 03 − − Frequency0.9999 1 1.0001 − × N u m be r o f S equen c e s Entries 1000Mean 03 − − Figure 9:
Frequencies of all combinations of length3 (000–999) in the decimal representation of the arclength of a lemniscate with a = 1 . Entries 16Mean 02 − − Frequency62.499 62.5 62.501 − × N u m be r o f S equen c e s Entries 16Mean 02 − − Figure 10:
Frequencies of all combinations of length1 (digits 0–9) in the hexadecimal representation of thearc length of a lemniscate with a = 1 . Entries 256Mean 03 − − Frequency3.906 3.9062 3.9064 3.9066 − × N u m be r o f S equen c e s Entries 256Mean 03 − − Figure 11:
Frequencies of all combinations of length2 (00–99) in the hexadecimal representation of the arclength of a lemniscate with a = 1 . Entries 4096Mean 04 − − Frequency0.2441 0.24415 0.2442 − × N u m be r o f S equen c e s Entries 4096Mean 04 − − Figure 12:
Frequencies of all combinations of length3 (000–999) in the hexadecimal representation of thearc length of a lemniscate with a = 1 .4im, S. Table 2:
Predicted and actual variances of frequencies of all sequences of length 1–3 in the decimal and hexadecimalrepresentations of the arc length of a lemniscate with a = 1 .Base Length of Sequence Predicted Variance and Error ofFrequencies Actual Variance of Frequencies Deviation [ σ ]10 1 (1 . ± . × − . × − − .
10 2 (1 . ± . × − . × − − .
10 3 (1 . ± . × − . × − − .
16 1 (1 . ± . × − . × − − .
16 2 (7 . ± . × − . × − − .
16 3 (4 . ± . × − . × − − . The frequencies of sequences from length 1 to 3 for both decimal and hexadecimal representations in both Catalan’sconstant and the arc length of a lemniscate with a = 1 overall coincide to the expected distribution. The distributionof frequencies for Catalan’s constant has a maximum deviation of . standard deviations (absolute value) forthe decimal representation, and . standard deviations (absolute value) for the hexadecimal representation. Thedistribution of frequencies for the arc length of a lemniscate with a = 1 has a maximum deviation of . standarddeviations (absolute value) for the decimal representation, and . standard deviations (absolute value) for thehexadecimal representation.These results for a very large sample data of around 600 billion digits of both Catalan’s constant and the arc lengthof a lemniscate with a = 1 are statistically persistent to the conjecture of Catalan’s constant and the arc length of alemniscate with a = 1 being a normal number in bases 10 and 16.In this study, it has been possible to make a statistical assumption of whether Catalan’s constant and the arc length of alemniscate with a = 1 is a normal number. Further approaches based on the methods used in this study can enablethe mathematical society to gain evidence on the normality of a large range of computable mathematical constants ofinterest. Acknowledgment
The author greatly thanks Dr. Peter Trueb for providing code and insights to formulating the methods for data analysisin this study, Mr. Alexander J. Yee for providing insights related to his program y-cruncher , and Dr. Ian Cutress forverifying the world record computation for the arc length of a lemniscate with a = 1 . Data Availability Statement
The data that support the findings of this study are openly available in the Internet Archive at https://archive.org/details/catalan_190618 , reference number ark:/13960/t9f55d078 , and at https://archive.org/details/lemworldrec_190512 , reference number ark:/13960/t56f3gj32 . References
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