In number theory, a perfect digital invariant (PDI) is a number in a given number base () that is the sum of its own digits each raised to a given power ().
Let be a natural number. The perfect digital invariant function (also known as a happy function, from happy numbers) for base and power is defined as:
where is the number of digits in the number in base , and
is the value of each digit of the number. A natural number is a perfect digital invariant if it is a fixed point for , which occurs if . and are trivial perfect digital invariants for all and , all other perfect digital invariants are nontrivial perfect digital invariants.
For example, the number 4150 in base is a perfect digital invariant with , because .
A natural number is a sociable digital invariant if it is a periodic point for , where for a positive integer (here is the th iterate of ), and forms a cycle of period . A perfect digital invariant is a sociable digital invariant with , and a amicable digital invariant is a sociable digital invariant with .
All natural numbers are preperiodic points for , regardless of the base. This is because if , , so any will satisfy until . There are a finite number of natural numbers less than , so the number is guaranteed to reach a periodic point or a fixed point less than , making it a preperiodic point.
Numbers in base lead to fixed or periodic points of numbers .
The number of iterations needed for to reach a fixed point is the perfect digital invariant function's persistence of , and undefined if it never reaches a fixed point.
is the digit sum. The only perfect digital invariants are the single-digit numbers in base , and there are no periodic points with prime period greater than 1.
reduces to , as for any power , and .
For every natural number , if , and , then for every natural number , if , then , where is Euler's totient function.
No upper bound can be determined for the size of perfect digital invariants in a given base and arbitrary power, and it is not currently known whether or not the number of perfect digital invariants for an arbitrary base is finite or infinite.
By definition, any three-digit perfect digital invariant for with natural number digits , , has to satisfy the cubic Diophantine equation . has to be equal to 0 or 1 for any , because the maximum value can take is . As a result, there are actually two related quadratic Diophantine equations to solve:
The two-digit natural number is a perfect digital invariant in base
This can be proven by taking the first case, where , and solving for . This means that for some values of and , is not a perfect digital invariant in any base, as is not a divisor of . Moreover, , because if or , then , which contradicts the earlier statement that .
There are no three-digit perfect digital invariants for , which can be proven by taking the second case, where , and letting and . Then the Diophantine equation for the three-digit perfect digital invariant becomes
for all values of . Thus, there are no solutions to the Diophantine equation, and there are no three-digit perfect digital invariants for .
By definition, any four-digit perfect digital invariant for with natural number digits , , , has to satisfy the quartic Diophantine equation . has to be equal to 0, 1, 2 for any , because the maximum value can take is . As a result, there are actually three related cubic Diophantine equations to solve
We take the first case, where .
Let be a positive integer and the number base . Then:
Let be a positive integer and the number base . Then:
Let be a positive integer and the number base . Then:
All numbers are represented in base .
Perfect digital invariants can be extended to the negative integers by use of a signed-digit representation to represent each integer.
In balanced ternary, the digits are 1, âÂÂ1 and 0. This results in the following:
A happy number for a given base and a given power is a preperiodic point for the perfect digital invariant function such that the -th iteration of is equal to the trivial perfect digital invariant , and an unhappy number is one such that there exists no such .
The example below implements the perfect digital invariant function described in the definition above to search for perfect digital invariants and cycles in Python. This can be used to find happy numbers.