### The Generalized Serial Test Applied to Expansions of Some Irrational Square Roots in Various Bases

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A brief summary is given of the application of the generalized serial test for randomness to the digits of irrational VIn in bases t where 2 < n, t < 15. The results are consistent, except for a few aberrations, with the hypothesis of randomness of the digits. This is a brief report on the application of the generalized serial test (see (1)) for randomness to the expansions of some irrational -\n in various bases. It can be con- sidered as an extension of the work Good and Gover (1) have done with 10,000 binary digits of V/2. I. J. Good (private communication) has remarked that some of the tests used in (4) are special cases of the generalized serial test. The conclusion resulting from this study is that the results, except for a few aberrations given in Table 3, are consistent with the hypothesis of randomness of the digits of the square roots investigated. Since a total of about 2400 tests were made, perhaps the aberrations are not surprising, if one notes that 2400-1 = .0004. For completeness, a recapitulation is given (from (1)) of the generalized serial test for randomness as used here. Let a sequence of N digits in base t (t = 2, 3, * * * ) be given and let the sequence be circularized; i.e., the last digit is considered as being followed by the first digit. Let nI be the number of occurrences of the v-plet I- (i4, i2, * , iv) in the circularized sequence. Define

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

[2] G. G. Stokes. "J." , 1890, The New Yale Book of Quotations.

[3] W. A. Beyer,et al. Statistical Study of Digits of Some Square Roots of Integers in Various Bases , 1970 .

[4] W. A. Beyer,et al. Square Roots of Integers 2 to 15 in Various Bases 2 to 10: 88062 Binary Digits or Equivalent , 1969 .