Happy number

A happy number is defined by the following process. Starting with any positive integer, replace the number by the sum of the squares of its digits, 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.

Overview

More formally, given a number $n=n_{0}$, define a sequence $n_{1}$, $n_{2}$, ... where $n_{i+1}$ is the sum of the squares of the digits of $n_{i}$. Then $n$ is happy if and only if there exists i such that $n_{i}=1$.

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, 7 is happy, as the associated sequence is:

7² = 49

4² + 9² = 97

9² + 7² = 130

1² + 3² + 0² = 10

1² + 0² = 1.

The happy numbers below 500 are

1, 7, 10, 13, 19, 23, 28, 31, 32, 44, 49, 68, 70, 79, 82, 86, 91, 94, 97, 100, 103, 109, 129, 130, 133, 139, 167, 176, 188, 190, 192, 193, 203, 208, 219, 226, 230, 236, 239, 262, 263, 280, 291, 293, 301, 302, 310, 313, 319, 320, 326, 329, 331, 338, 356, 362, 365, 367, 368, 376, 379, 383, 386, 391, 392, 397, 404, 409, 440, 446, 464, 469, 478, 487, 490, 496. (sequence A007770 in the OEIS).

Sequence behavior

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 $81m$. For $m=4$ and above,

$n\geq 10^{m-1}>81m$

so any number over 1000 gets smaller under this process. 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.

Considering more precisely the intervals [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] is either happy or goes to the above cycle.

Happy primes

A happy prime is a happy number that is prime. The happy primes below 500 are

7, 13, 19, 23, 31, 79, 97, 103, 109, 139, 167, 193, 239, 263, 293, 313, 331, 367, 379, 383, 397, 409, 487 (sequence A035497 in the OEIS).

All numbers, and therefore all primes, of the form $10^{n}+3$ and $10^{n}+9$ are happy. To see this, note that these numbers yield values of either 1² + 0² + 0² + ... + 0² + 0² + 3² = 10 → 1² + 0² = 1 or 1² + 0² + 0² + ... + 0² + 0² + 9² = 82 → 8² + 2² = 68 → 6² + 8² = 100 → 1² + 0² + 0² = 1.

The palindromic prime 10¹⁵⁰⁰⁰⁶ + 7426247×10⁷⁵⁰⁰⁰ + 1 is also a happy prime with 150007 digits because the many 0's do not contribute to the sum of squared digits, and $1^{2}+7^{2}+4^{2}+2^{2}+6^{2}+2^{2}+4^{2}+7^{2}+1^{2}=176$, which is a happy number. Paul Jobling discovered the prime in 2005..

As of June 2007, the largest known happy prime and the twelfth largest known prime is 4847 × 2^3321063 + 1. The decimal expansion has 999744 digits: 1844857508...(999724 digits omitted)...2886501377. Richard Hassler and Seventeen or Bust discovered the prime in 2005. Jens K. Andersen identified it as the largest known happy prime in June 2007.

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, $100_{2}$ represents the number 4, and

$123_{5}=1\cdot 5^{2}+2\cdot 5+3=38.$

Then, it is easy to see that there are happy numbers in every base. For instance, the numbers

$1_{b},10_{b},100_{b},1000_{b},...$

are all happy, for any base $b$.

By a similar argument to the one above for decimal happy numbers, we can see that unhappy numbers in base $b$ lead to cycles of numbers less than $1000_{b}$. We can use the fact that if $n<1000_{b}$, then the sum of the squares of the base-$b$ digits of $n$ is less than or equal to

$3(b-1)^{2}$

which can be shown to be less than $b^{3}$. This shows that once the sequence reaches a number less than $1000_{b}$, it stays below $1000_{b}$, 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 $1000_{2}$:

$111_{2}\rightarrow 11_{2}\rightarrow 10_{2}\rightarrow 1$

$110_{2}\rightarrow 10_{2}\rightarrow 1$

$101_{2}\rightarrow 10_{2}\rightarrow 1$

$100_{2}\rightarrow 1.$

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 happy bases strictly less than 2400 are 2 and 4, yet more could exist outside this range. This result may easily be verified and improved upon by running the following code in Mathematica:

x=1;h={};i=0;Label[j];x++;Print[x];t=2;Label[k];If[t==x^2-1,Print[x];Print["Happy!"];i++;AppendTo[h,x];Goto[j],];t++;c=t;y=0;l={};Label[m];a=Floor[c/x^2];c=c-a*x^2;b=Floor[c/x];c=c-b*x;c=a^2+b^2+c^2;If[c==1,Goto[k],];y++;AppendTo[l,c];If[y>2*x,For[z=1,z<y,z++,If[c==lz,Goto[j],]],];Goto[m]

Origin

Happy numbers were brought to the attention of Reg Allenby[1], a British author and Senior Lecturer in pure mathematics at Leeds University, by his daughter. She had learned of them at school, but they "may have originated in Russia" (Guy 2004:§E34).

Popular culture

In the Doctor Who episode "42", a sequence of happy primes (313, 331, 367, 379) is used as a code for unlocking a sealed door on a spaceship about to collide with a sun.

References

• Walter Schneider, Mathews: Happy Numbers.
• Weisstein, Eric W. "Happy Number". MathWorld.
• Happy Numbers at The Math Forum.

• Guy, Richard (2004), Unsolved Problems in Number Theory (third edition), Springer-Verlag, ISBN 0-387-20860-7