Happy Numbers

Begin with a positive whole number written in base 10. Multiply each digit by itself, add those squares and use the total as the next number. For example, 19 gives 1² + 9² = 82; then 82 gives 8² + 2² = 68.

Repeating the operation gives 68 → 100 → 1. Since 1² = 1, reaching 1 means every later value is also 1. A starting number with this outcome is called happy.

For this base-10 square-and-add rule, every positive start either reaches 1 or enters 4 → 16 → 37 → 58 → 89 → 145 → 42 → 20 → 4. Those are the only two outcomes, as the finite reduction below proves. Changing the base or the power changes the rule, so that two-outcome statement should not be carried over to the other settings.

The rule has a ceiling, and the ceiling is very low.

A four-digit number is at least 1000. The most its digits can contribute is four nines, which is $4\times 81=324$. So every four-digit number gets smaller. The same is true of five digits, six digits, and everything above - the number grows by a factor of ten per digit while its image grows by only 81.

So no orbit can run away. Whatever you start with, within a few steps you are below 1000 and you stay below 1000. And once a rule is confined to a thousand values and never stops applying, it has to revisit something. From then on it repeats forever.

That is the whole reason this page can be certain of anything: the interesting behaviour has nowhere to hide.

Write $S_p(n)$ for the sum of the $p$-th powers of the digits of $n$ in base $b$. If $n$ has $d$ digits then

The left-hand quantity is exponential in $d$; the right-hand one is linear. So there is a width $D$ at which the exponential wins:

Every number with $D$ digits or more therefore satisfies $S_p(n)<n$ and strictly decreases. And any number below $b^{D-1}$ has at most $D-1$ digits, so its image is at most $(D-1)(b-1{)}^p$, which is smaller still - the interval $[1,b^{D-1}-1]$ maps into itself.

For base 10 and squares, $D=4$: the interval is $1\dots 999$.

A map from a finite set to itself, applied forever, must repeat. The dynamics are a finite functional graph, and this page enumerates it.

Fix a base $b\ge 2$ and a power $p\ge 2$, and let $S$ be the digit-power map.

1. There is an explicit $R=R(b,p)$ with $S([1,R])\subseteq [1,R]$, and for every $n>R$, $S(n)<n$.
2. Consequently every orbit enters $[1,R]$ and is eventually periodic.
3. The cycles of $S$ are exactly the cycles of $S$ restricted to $[1,R]$ - a finite list, which the Destinations view shows in full.
4. For $b=10$, $p=2$: $R=999$, and the list is $\{1\}$ together with $\{4,16,37,58,89,145,42,20\}$.

Statement 4 is a theorem, not a report of a search. Statements 1 to 3 are what make it one.

The bound. As in layer 3. Both inequalities are immediate from the definition of base-$b$ notation, and the comparison $D(b-1{)}^p<b^{D-1}$ holds for all larger widths too: increasing $D$ by one multiplies the right-hand side by $b$ and the left-hand side by $\frac{D+1}{D}\le \frac{3}{2}$, and $b\ge 2$.

Eventual periodicity. Orbits are non-increasing once above $R$ and confined once below, so after finitely many steps the orbit lies in a set of $R$ elements. An infinite sequence in a finite set repeats a value; from the first repeat onward the orbit is periodic, because $S$ is a function of the current value alone.

The cycle list. Every cycle lies in $[1,R]$, and $[1,R]$ has been enumerated. Nothing outside it can be periodic, because points outside strictly decrease. ∎

The last line is the part worth pausing on. "We checked all of them" is normally a confession; here it is the end of the proof, because a separate argument established that all of them is a finite list.

What is settled is settled. For any base and power in reach, the cycles, the basin sizes and the longest journey to 1 are complete answers. There is no larger computation that would refine them.

What is not settled is how many. Look at the "How many are happy" view. The proportion of happy numbers below $x$ drops from 30% to 20% to 14.3% and then appears to hold there. It looks exactly like a sequence converging to a density.

There is no density. The upper limit of that proportion has been proved to be strictly greater than its lower limit, so the ratio keeps moving and never settles. The oscillation happens at scales no browser will ever draw. Everything visible in that figure is consistent with a convergence that does not exist - which is a good reason to distrust flat-looking tails in general.

The refused cells are real. In the Base × power view, six combinations are hatched out. Their regions are finite - that is a theorem and it applies to every base and power - but finite is not the same as small: base 16 with fourth powers confines its orbits to about a million states, and this page declines to enumerate them. The claim "the space is finite" and the claim "we can check the space" are different claims, and conflating them is how a lot of computational mathematics goes wrong.

Compare with Gilbreath's difference field. There, a verification to $10^{13}$ establishes nothing about the conjecture, because the space is infinite and $10^{13}$ is a prefix of it. Here, a verification of 999 states establishes everything. The difference is not diligence. It is whether anything forces the space to close.