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:
“One should never direct people towards happiness, because happiness too is an idol of the market-place. One should direct them towards mutual affection. A beast gnawing at its prey can be happy too, but only human beings can feel affection for each other, and this is the highest achievement they can aspire to.”
—Alexander Solzhenitsyn (b. 1918)
“I had a feeling that out there, there were very poor people who didn’t have enough to eat. But they wore wonderfully colored rags and did musical numbers up and down the streets together.”
—Jill Robinson (b. 1936)
“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)