Schizophrenic number

Definition

The definition of schizophrenic numbers is given in The Universal Book of Mathematics as:

An informal name for an irrational number that displays such persistent patterns in its decimal expansion, that it has the appearance of a rational number. A schizophrenic number can be obtained as follows. For any positive integer n let f(n) denote the integer given by the recurrence f(n) = 10 f(n − 1) + n with the initial value f(0) = 0. Thus, f(1) = 1, f(2) = 12, f(3) = 123, and so on. The square roots of f(n) for odd integers n give rise to a curious mixture appearing to be rational for periods, and then disintegrating into irrationality. This is illustrated by the first 500 digits of √f(49): The repeating strings become progressively shorter and the scrambled strings become larger until eventually the repeating strings disappear. However, by increasing n we can forestall the disappearance of the repeating strings as long as we like. The repeating digits are always 1, 5, 6, 2, 4, 9, 6, 3, 9, 2,....

The sequence of numbers generated by the recurrence relation f(n) = 10 f(n − 1) + n described above is:

0, 1, 12, 123, 1234, 12345, 123456, 1234567, 12345678, 123456789, 1234567900, ... (sequence A014824 in the OEIS).

The integer parts of their square roots,

1, 3, 11, 35, 111, 351, 1111, 3513, 11111, 35136, 111111, 351364, 1111111, ... (sequence A068995 in the OEIS),

alternate between numbers with irregular digits and numbers with repeating digits, in a similar way to the alternations appearing within the fractional part of each square root.

History

Clifford A. Pickover has said that the schizophrenic numbers were discovered by Kevin Brown.

In his book Wonders of Numbers he has so described the history of schizophrenic numbers:

The construction and discovery of schizophrenic numbers was prompted by a claim (posted in the Usenet newsgroup sci.math) that the digits of an irrational number chosen at random would not be expected to display obvious patterns in the first 100 digits. It was said that if such a pattern were found, it would be irrefutable proof of the existence of either God or extraterrestrial intelligence. (An irrational number is any number that cannot be expressed as a ratio of two integers. Transcendental numbers like e and π, and noninteger surds such as square root of 2 are irrational.)