Happy Numbers in Other Bases
The definition of happy numbers depends on the decimal (i.e., base 10) representation of the numbers. The definition can be extended to other bases.
To represent numbers in other bases, we may use a subscript to the right to indicate the base. For instance, represents the number 4, and
Then, it is easy to see that there are happy numbers in every base. For instance, the numbers
are all happy, for any base b.
By a similar argument to the one above for decimal happy numbers, unhappy numbers in base b lead to cycles of numbers less than . If, then the sum of the squares of the base-b digits of n is less than or equal to
which can be shown to be less than . This shows that once the sequence reaches a number less than, it stays below, and hence must cycle or reach 1.
In base 2, all numbers are happy. All binary numbers larger than 10002 decay into a value equal to or less than 10002, and all such values are happy: The following four sequences contain all numbers less than :
Since all sequences end in 1, we conclude that all numbers are happy in base 2. This makes base 2 a happy base.
The only known happy bases are 2 and 4. There are no others less than 500,000,000.
Read more about this topic: Happy Number
Famous quotes containing the words happy, numbers and/or bases:
“All who wish to hand down to their children that happy republican system bequeathed to them by their revolutionary fathers, must now take their stand against this consolidating, corrupting money power, and put it down, or their children will become hewers of wood and drawers of water to this aristocratic ragocracy.”
—Andrew Jackson (1767–1845)
“The barriers of conventionality have been raised so high, and so strangely cemented by long existence, that the only hope of overthrowing them exists in the union of numbers linked together by common opinion and effort ... the united watchword of thousands would strike at the foundation of the false system and annihilate it.”
—Mme. Ellen Louise Demorest 1824–1898, U.S. women’s magazine editor and woman’s club movement pioneer. Demorest’s Illustrated Monthly and Mirror of Fashions, p. 203 (January 1870)
“The information links are like nerves that pervade and help to animate the human organism. The sensors and monitors are analogous to the human senses that put us in touch with the world. Data bases correspond to memory; the information processors perform the function of human reasoning and comprehension. Once the postmodern infrastructure is reasonably integrated, it will greatly exceed human intelligence in reach, acuity, capacity, and precision.”
—Albert Borgman, U.S. educator, author. Crossing the Postmodern Divide, ch. 4, University of Chicago Press (1992)