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:
“Dear, why should you command me to my rest,
When now the night doth summon all to sleep?
Methinks this time becometh lovers best;
Night was ordained together friends to keep.
How happy are all other living things,
Which though the day disjoin by several flight,
The quiet evening yet together brings,
And each returns unto his love at night.”
—Michael Drayton (1563–1631)
“One murder makes a villain, millions a hero. Numbers sanctify, my good fellow.”
—Charlie Chaplin (1889–1977)
“In the beginning was the word, the word
That from the solid bases of the light
Abstracted all the letters of the void....”
—Dylan Thomas (1914–1953)