A happy number is defined by the following process. Starting with any positive integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
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A digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...
, and repeat the process until the number equals 1 (where it will stay), or it loops endlessly in a cycle which does not include 1. Those numbers for which this process ends in 1 are happy numbers, while those that do not end in 1 are unhappy numbers (or sad numbers).
Overview
More formally, given a number , define a sequence , , ... where is the sum of the squares of the digits of . Then n is happy if and only if there exists i such that .
If a number is happy, then all members of its sequence are happy; if a number is unhappy, all members of its sequence are unhappy.
For example, 19 is happy, as the associated sequence is:
If n is not happy, then its sequence does not go to 1. What happens instead is that it ends up in the cycle
4, 16, 37, 58, 89, 145, 42, 20, 4, ...
To see this fact, first note that if n has m digits, then the sum of the squares of its digits is at most , or .
For and above,
so any number over 1000 gets smaller under this process and in particular becomes a number with strictly fewer digits. Once we are under 1000, the number for which the sum of squares of digits is largest is 999, and the result is 3 times 81, that is, 243.
In the range 100 to 243, the number 199 produces the largest next value, of 163.
In the range 100 to 163, the number 159 produces the largest next value, of 107.
In the range 100 to 107, the number 107 produces the largest next value, of 50.
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
[244,999], [164,243], [108,163] and [100,107], we see that every number above 99 gets strictly smaller under this process. Thus, no matter what number we start with, we eventually drop below 100. An exhaustive search then shows that every number in the interval [1,99] either is happy or goes to the above cycle.
The above work produces the interesting result that no positive integer other than 1 is the sum of the squares of its own digits, since any such number would be a fixed point of the described process.
There are infinitely many happy numbers and infinitely many unhappy numbers. For example, if one wonders if any number will produce 14,308, say, the quick response is to write down the digit "1" 14,308 times and you have created such a number. In fact, infinitely many such numbers exist since zeros can be inserted at will.
The first pair of consecutive happy numbers is 31, 32. The first set of triplets is 1880, 1881, and 1882.
An interesting question is to wonder about the density of happy numbers. In the interval , 15.5% (to three significant figures) are happy.
Happy primes
A happy prime is a number that is both happy and prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
All numbers, and therefore all primes, of the form 10n + 3 and 10n + 9 for n greater than 0 are Happy (This of course does not mean that these are the only happy primes, as evidenced by the sequence above). To see this, note that
All such numbers will have at least 2 digits;
The first digit will always be 1 due to the 10n
The last digit will always be either 3 or 9.
Any other digits will always be 0 (and therefore will not contribute to the sum of squares of the digits).
The sequence for adding 3 is: 12 + 32 = 10 → 12 = 1
A palindromic prime is a prime number that is also a palindromic number. Palindromicity depends on the base of the numbering system and its writing conventions, while primality is independent of such concerns...
10150006 + 7426247 + 1 is also a happy prime with 150,007 digits because the many 0's do not contribute to the sum of squared digits, and , which is a happy number. Paul Jobling discovered the prime in 2005.
In mathematics, a Mersenne number, named after Marin Mersenne , is a positive integer that is one less than a power of two: M_p=2^p-1.\,...
). Its decimal expansion has 12,837,064 digits.
Special happy numbers
986,543,210 : Greatest happy number with no redundant digits
Numerical digit
A digit is a symbol used in combinations to represent numbers in positional numeral systems. The name "digit" comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 number system, i.e...
In mathematics, a pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. For example, 1223334444555567890 is a pandigital number in base 10...
A palindromic number or numeral palindrome is a 'symmetrical' number like 16461, that remains the same when its digits are reversed. The term palindromic is derived from palindrome, which refers to a word like rotor that remains unchanged under reversal of its letters...
happy number
1,034,567,892,987,654,301 : Smallest pandigital palindromic happy number
A Pythagorean triple consists of three positive integers a, b, and c, such that . Such a triple is commonly written , and a well-known example is . If is a Pythagorean triple, then so is for any positive integer k. A primitive Pythagorean triple is one in which a, b and c are pairwise coprime...
with all integers happy and less than 10000
( 700, 3465, 3535)
( 748, 8211, 8245)
( 910, 8256, 8306)
( 940, 2109, 2309)
( 940, 4653, 4747)
(1092, 1881, 2175)
(1323, 4536, 4725)
(1527, 2036, 2545)
(1785, 3392, 3833)
(1900, 1995, 2755)
(1995, 4788, 5187)
(2715, 3620, 4525)
(2751, 8360, 8801)
(2784, 6440, 7016)
(3132, 7245, 7893)
(3135, 7524, 8151)
(3290, 7896, 8554)
(3367, 3456, 4825)
(3680, 5313, 6463)
(4284, 5313, 6825)
(4633, 5544, 7225)
(5178, 6904, 8630)
(5286, 7048, 8810)
(5445, 6308, 8333)
(5712, 7084, 9100)
(6528, 7480, 9928)
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
Radix
In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system the radix is ten, because it uses the ten digits from 0 through 9.In any numeral...
.
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.
The binary numeral system, or base-2 number system, represents numeric values using two symbols, 0 and 1. More specifically, the usual base-2 system is a positional notation with a radix of 2...
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.
Cubing the digits rather than squaring
An interesting extension to the Happy Numbers problem is to find the sum of the cubes of the digits rather than the sum of the squares of the digits. For example, working in base 10, 1579 is happy, since:
13+53+73+93=1+125+343+729=1198
13+13+93+83=1+1+729+512=1243
13+23+43+33=1+8+64+27=100
13+03+03=1
In the same way that when summing the squares of the digits (and working in base 10) each number above 243(=3*81) produces a number which is strictly smaller, when summing the cubes of the digits each number above 2916(=4*729) produces a number which is strictly smaller.
By conducting an exhaustive search of [1,2916] one finds that for summing the cubes of digits base 10 there are happy numbers and eight different types of unhappy number:
those that eventually reach which perpetually produces itself.
those that eventually reach which perpetually produces itself.
those that eventually reach the loop
those that eventually reach which perpetually produces itself.
those that eventually reach the loop
those that eventually reach which perpetually produces itself.
those that eventually reach the loop
those that eventually reach the loop
Starting with the happy numbers and then following with the unhappy numbers in the order given above, the density of each type of number in the interval [1,2916] is 1.54%, 28.4%, 34.7%, 5.73%, 17.4%, 4.60%, 3.60%, 2.67% and 1.34% (all to 3 significant figures).
Intriguingly, the second type of unhappy number includes all multiples of three . This fact can be proved by the exhaustive search up to 2916 and noting that a number is a multiple of three if and only if the sum of digits is a multiple of three if and only if the sum of its cubed digits are a multiple of three. By similar reasoning, all happy numbers of this type must have a remainder of 1 when dividing by 3.
One interesting result which comes from the above work is that the only positive whole numbers which are the sum of the cubes of their digits are 1, 153, 370, 371 and 407.
Origin
The origin of happy numbers is not clear. Happy numbers were brought to the attention of Reg Allenby (a British author and Senior Lecturer in pure mathematics
Pure mathematics
Broadly speaking, pure mathematics is mathematics which studies entirely abstract concepts. From the eighteenth century onwards, this was a recognized category of mathematical activity, sometimes characterized as speculative mathematics, and at variance with the trend towards meeting the needs of...
at Leeds University) by his daughter, who had learned of them at school. However, they "may have originated in Russia" .
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"42" is an episode of the British science fiction television series Doctor Who. It was first broadcast on BBC One on 19 May 2007, and is the seventh episode of Series 3 of the revived Doctor Who series....
", a sequence of happy primes (313, 331, 367, 379) is used as a code
Code
A code is a rule for converting a piece of information into another form or representation , not necessarily of the same type....
for unlocking a sealed door on a spaceship about to collide with a sun. When the Doctor learns that nobody on the spaceship besides himself has heard of happy numbers, he asks, "Don't they teach recreational mathematics anymore?"
Programming example
The examples below apply the 'happy' process described in the definition of happy given at the top of this article, repeatedly; after each time, they check for both halt conditions: reaching 1, and repeating a number. Everything else is book-keeping (for example, the Python example precomputes the squares of all 10 digits.
Python is a general-purpose, high-level programming language whose design philosophy emphasizes code readability. Python claims to "[combine] remarkable power with very clear syntax", and its standard library is large and comprehensive...
to check if a number is happy.
SQUARE = dict([(c, int(c) ** 2) for c in "0123456789"])
def is_happy(n):
s = set
while (n > 1) and (n not in s):
s.add(n)
n = sum(SQUARE[d] for d in str(n))
return n 1
, define a sequence 
, 
, ... where 
is the sum of the squares of the digits of 
. Then n is happy if and only if there exists i such that 
.
, or 
.
and above,
, 15.5% (to three significant figures) are happy.
, which is a happy number. Paul Jobling discovered the prime in 2005.
(Mersenne prime
represents the number 4, and
. If 
, then the sum of the squares of the base-b digits of n is less than or equal to
. This shows that once the sequence reaches a number less than 
, it stays below 
, and hence must cycle or reach 1.
:
which perpetually produces itself.
which perpetually produces itself.
which perpetually produces itself.
which perpetually produces itself.
