Shifting Digits and a Point of View
The number xn is defined as the last digit in the decimal representation of the integer ⌊(√2)n⌋ (n = 1, 2, ...). Determine whether the sequence x1, x2, ..., xn, ... is periodic. [Savchev & Andreescu, p. 146].
(⌊a⌋ is the whole part of number a which is more often at this site and elsewhere denoted [x].)
Recollect that in the decimal system the multiplication of a real number by 10 causes the decimal point (or comma in some cultures) to shift one position to the right. The same happens in the binary system when a number is multiplied by 2. It follows that y2n+1 is a digit in the binary representation of √2 so that 0.y1y3y5... is the binary representation of the fractional part of √2. Since √2 is irrational, the sequence y1, y3, y5, ... is not periodic.
Let xn = zn (mod 2), where zn is either 0 or 1, n = 1, 2, ... Clearly,
- S. Savchev, T. Andreescu, Mathematical Miniatures, MAA, 2003
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