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

Abstract

Some statistical tests of randomness are made of the first 88062 binary digits (or equivalent in other bases) of √ n \surd n in various bases b b , 2 ≦ n ≦ 15 2 \leqq n \leqq 15 ( n n square-free) with b = 2 , 4 , 8 , 16 b = 2,4,8,16 and n = 2 , 3 , 5 n = 2,3,5 with b = 3 , 5 , 6 , 7 b = 3,5,6,7 , and 10 10 . The statistical tests are the χ 2 {\chi ^2} 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 √ n \surd n in the assigned bases b 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).