Statistical Study of Digits of Some Square Roots of Integers in Various Bases

Some statistical tests of randomness are made of the first 88062 binary digits (or equiva- lent in other bases) of In in various bases b, 2 < n < 15 (n square-free) with b = 2, 4, 8, 16 and n = 2, 3, 5 with b = 3, 5, 6, 7, and 10. The statistical tests are the x2 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 random- ness or normality appears for the digits of the above mentioned ,/n in the assigned bases b. 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]  W. A. Beyer,et al.  Square Roots of Integers 2 to 15 in Various Bases 2 to 10: 88062 Binary Digits or Equivalent , 1969 .

[2]  W. A. Beyer,et al.  Clusters on a Thin Quadratic Lattice (Transfer Matrix Technique) , 1969 .

[3]  W. F. Lunnon,et al.  Expansion of √2 to 100,000 Decimals , 1968 .

[4]  M. F. Jones.  22900D Approximation to the Square Roots of the Primes Less than 100 , 1968 .

[5]  H. Ko.  An introduction to probability theory and its applications, Vol. II: by William Feller. 626 pages, 6 × 9 inches, New York, John Wiley and Sons, Inc., 1966. Price $12.00 , 1967 .

[6]  M. Sibuya,et al.  The Decimal and Octal Digits of √n , 1967 .

[7]  I. Good,et al.  Corrigendum: The Generalized Serial Test and the Binary Expansion of √2 , 1967 .

[8]  Arthur H. Kruse,et al.  Some Notions of Random Sequence and Their Set-Theoretic Foundations , 1967 .

[9]  Per Martin-Löf,et al.  The Definition of Random Sequences , 1966, Inf. Control..

[10]  Harald Bergstriim.  Mathematical Theory of Probability and Statistics , 1966 .

[11]  R. G. Stoneham.  A Study of 60,000 Digits of the Transcendental “e” , 1965 .

[12]  Richard Von Mises,et al.  Mathematical Theory of Probability and Statistics , 1966 .

[13]  W. Feller.  An Introduction to Probability Theory and Its Applications , 1959 .

[14]  N. Metropolis,et al.  Statistical treatment of values of first 2,000 decimal digits of and of calculated on the ENIAC , 1950 .

[15]  George W. Reitwiesner.  An ENIAC Determination of π and e to more than 2000 Decimal Places , 1950 .

[16]  P. Erdös,et al.  On the number of positive sums of independent random variables , 1947 .

[17]  S. S. Pillai,et al.  On normal numbers , 1939 .

[18]  F. Yates,et al.  Statistical Tables for Biological, Agricultural and Medical Research. , 1939 .

[19]  W. E. H. B.,et al.  Aufgaben und Lehrsätze aus der Analysis. , 1925, Nature.