In number theory, a perfect digital invariant (PDI) is a number in a given number base (${\displaystyle b}$) that is the sum of its own digits each raised to a given power (${\displaystyle p}$).

Let ${\displaystyle n}$ be a natural number. The perfect digital invariant function (also known as a happy function, from happy numbers) for base ${\displaystyle b>1}$ and power ${\displaystyle p>0}$ ${\displaystyle F_{p,b}:\mathbb {N} \rightarrow \mathbb {N} }$ is defined as:

where ${\displaystyle k=\lfloor \log _{b}{n}\rfloor +1}$ is the number of digits in the number in base ${\displaystyle b}$, and

is the value of each digit of the number. A natural number ${\displaystyle n}$ is a perfect digital invariant if it is a fixed point for ${\displaystyle F_{p,b}}$, which occurs if ${\displaystyle F_{p,b}(n)=n}$. ${\displaystyle 0}$ and ${\displaystyle 1}$ are trivial perfect digital invariants for all ${\displaystyle b}$ and ${\displaystyle p}$, all other perfect digital invariants are nontrivial perfect digital invariants.

For example, the number 4150 in base ${\displaystyle b=10}$ is a perfect digital invariant with ${\displaystyle p=5}$, because ${\displaystyle 4150=4^{5}+1^{5}+5^{5}+0^{5}}$.

A natural number ${\displaystyle n}$ is a sociable digital invariant if it is a periodic point for ${\displaystyle F_{p,b}}$, where ${\displaystyle F_{p,b}^{k}(n)=n}$ for a positive integer ${\displaystyle k}$ (here ${\displaystyle F_{p,b}^{k}}$ is the ${\displaystyle k}$th iterate of ${\displaystyle F_{p,b}}$), and forms a cycle of period ${\displaystyle k}$. A perfect digital invariant is a sociable digital invariant with ${\displaystyle k=1}$, and a amicable digital invariant is a sociable digital invariant with ${\displaystyle k=2}$.

All natural numbers ${\displaystyle n}$ are preperiodic points for ${\displaystyle F_{p,b}}$, regardless of the base. This is because if ${\displaystyle k\geq p+2}$, ${\displaystyle n\geq b^{k-1}>b^{p}k}$, so any ${\displaystyle n}$ will satisfy ${\displaystyle n>F_{b,p}(n)}$ until ${\displaystyle n<b^{p+1}}$. There are a finite number of natural numbers less than ${\displaystyle b^{p+1}}$, so the number is guaranteed to reach a periodic point or a fixed point less than ${\displaystyle b^{p+1}}$, making it a preperiodic point.

Numbers in base ${\displaystyle b>p}$ lead to fixed or periodic points of numbers ${\displaystyle n\leq (p-2)^{p}+p(b-1)^{p}}$.