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:
“When I was sick and lay a-bed,
I had two pillows at my head,
And all my toys beside me lay
To keep me happy all the day.”
—Robert Louis Stevenson (1850–1894)
“The only phenomenon with which writing has always been concomitant is the creation of cities and empires, that is the integration of large numbers of individuals into a political system, and their grading into castes or classes.... It seems to have favored the exploitation of human beings rather than their enlightenment.”
—Claude Lévi-Strauss (b. 1908)
“The bases for historical knowledge are not empirical facts but written texts, even if these texts masquerade in the guise of wars or revolutions.”
—Paul Deman (1919–1983)