Statistical study of digits of some square roots of integers in various bases

Authors:
W. A. Beyer, N. Metropolis and J. R. Neergaard

Journal:
Math. Comp. **24** (1970), 455-473

MSC:
Primary 62.70

DOI:
https://doi.org/10.1090/S0025-5718-1970-0272129-1

Corrigendum:
Math. Comp. **25** (1971), 409.

Corrigendum:
Math. Comp. **25** (1971), 409.

MathSciNet review:
0272129

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Abstract | References | Similar Articles | Additional Information

Abstract: Some statistical tests of randomness are made of the first 88062 binary digits (or equivalent in other bases) of in various bases , ( square-free) with and with , and . The statistical tests are the test for cumulative frequency distribution of the digits, the lead test, and the gap test. The lead test is an examination of the distances over which the cumulative frequency of a digit exceeded its expected value. It is related to the arc sine law. The gap test (applied to the binary digits) consists of an examination of the distribution of runs of ones. The conclusion of the study is that no evidence of the lack of randomness or normality appears for the digits of the above mentioned in the assigned bases . It seems to be the first statistical study of the digits of any naturally occurring number in bases other than decimal or binary (octal).

**[1]**É. Borel,*Probability and Certainty*, Walker, New York, 1963.**[2]**J. Marcus boorman, "Square-root notes,"*Math. Mag.*, v. 1, 1887, pp. 207-208.**[3]**Harald Cramér,*Mathematical Methods of Statistics*, Princeton Mathematical Series, vol. 9, Princeton University Press, Princeton, N. J., 1946. MR**0016588****[4]**Editor,*Math. Mag.*, v. 1, 1887, p. 164.**[5]**P. Erdös and M. Kac,*On the number of positive sums of independent random variables*, Bull. Amer. Math. Soc.**53**(1947), 1011–1020. MR**0023011**, https://doi.org/10.1090/S0002-9904-1947-08928-X**[6]**William Feller,*An introduction to probability theory and its applications. Vol. I*, Third edition, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR**0228020****[7]**Ronald A. Fisher and Frank Yates,*Statistical Tables for Biological, Agricultural and Medical Research*, Oliver and Boyd Ltd., London, 1943. 2nd ed. MR**0009818****[8]**I. J. Good, "The generalized serial test and the binary expansion of ,"*J. Roy. Statist. Soc Ser. A*, v. 130, 1967, pp. 102-107.**[9]**M. F. Jones, "Approximation to the square roots of primes less than 100,"*Math. Comp.*, v. 21, 1967, p. 234.**[10]**Arthur H. Kruse,*Some notions of random sequence and their set-theoretic foundations*, Z. Math. Logik Grundlagen Math.**13**(1967), 299–322. MR**0226685****[11]**M. Lal, "Expansion of to 19,600 decimals,"*Math. Comp.*, v. 21, 1967, p. 258.**[12]**M. Lal & W. F. Lunnon, "Expansion of to 100,000 Decimals,"*Math. Comp.*, v. 22, 1968, pp. 899-900.**[13]**Per Martin-Löf,*The definition of random sequences*, Information and Control**9**(1966), 602–619. MR**0223179****[14]**N. C. Metropolis, G. Reitwiesner, and J. von Neumann,*Statistical treatment of values of first 2,000 decimal digits of 𝑒 and of 𝜋 calculated on the ENIAC*, Math. Tables and Other Aids to Computation**4**(1950), 109–111. MR**0037598**, https://doi.org/10.1090/S0025-5718-1950-0037598-8**[15]**Richard von Mises,*Mathematical theory of probability and statistics*, Edited and Complemented by Hilda Geiringer, Academic Press, New York-London, 1964. MR**0178486****[16]**G. Pólya and G. Szegö,*Aufgaben und Lehrsätze aus der Analysis. Band I: Reihen. Integralrechnung. Funktionentheorie*, Dritte berichtigte Auflage. Die Grundlehren der Mathematischen Wissenschaften, Band 19, Springer-Verlag, Berlin-New York, 1964 (German). MR**0170985**

G. Pólya and G. Szegö,*Aufgaben und Lehrsätze aus der Analysis. Band II: Funktionentheorie. Nullstellen. Polynome. Determinanten. Zahlentheorie*, Dritte berichtigte Auflage. Die Grundlehren der Mathematischen Wissenschaften, Band 20, Springer-Verlag, Berlin-New York, 1964 (German). MR**0170986****[17]**George W. Reitwiesner,*An ENIAC determination of 𝜋 and 𝑒 to more than 2000 decimal places*, Math. Tables and Other Aids to Computation**4**(1950), 11–15. MR**0037597**, https://doi.org/10.1090/S0025-5718-1950-0037597-6**[18]**Wolfgang M. Schmidt,*On normal numbers*, Pacific J. Math.**10**(1960), 661–672. MR**0117212****[19]**R. G. Stoneham,*A study of 60,000 digits of the transcendental “𝑒”*, Amer. Math. Monthly**72**(1965), 483–500. MR**0179871**, https://doi.org/10.2307/2314118**[20]**Kōki Takahashi & Masaaki Sibuya, "The decimal and octal digits of ,"*MTAC*, v. 21, 1967, pp. 259-260.**[21]**Kōki Takahashi & Masaaki Sibuya, "Statistics of the digits of ,"*Joho Shori (Information Processing*), v. 6, 1965, pp. 221-223. (Japanese)**[22]**Horace S. Uhler, "Many-figure approximations to , and distribution of digits in and ,"*Proc. Nat. Acad. Sci. U.S.A.*, v. 37, 1951, pp. 63-67. MR**12**, 444.**[23]**Horace S. Uhler, "Approximations exceeding 1300 decimals for , , and distribution of digits in them,"*Proc. Nat. Acad. Sci. U.S.A.*, v. 37, 1951, pp. 443-447. MR**13**, 161.**[24]**W. A. Beyer, N. Metropolis & J. R. Neergaard, "Square roots of integers 2 to 15 in various bases 2 to 10: 88062 binary digits or equivalent,"*Math. Comp.*, v. 23, 1969, p. 679.**[25]**Elliott H. Lieb & W. A. Beyer, "Clusters on a thin quadratic lattice (transfer matrix technique),"*Studies in Appl. Math.*, v. 48, 1969, pp. 77-90.

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0272129-1

Keywords:
Statistics of square root digits,
square roots,
square-roots in several bases,
expansions of square roots,
random sequences,
statistical study of digit sequences,
radix transformations

Article copyright:
© Copyright 1970
American Mathematical Society