In number theory, the multiplicative digital root of a natural number
is found by multiplying the digits of
together, then repeating this operation until only a single-digit remains, which is called the multiplicative digital root of
[1][2] The multiplicative digital root for the first few positive integers are: Multiplicative digital roots are the multiplicative equivalent of digital roots.
We define the digit product for base
, and is the value of each digit of the number.
A natural number
is a multiplicative digital root if it is a fixed point for
, 0 is the multiplicative digital root of 9876, as All natural numbers
, then trivially Therefore, the only possible multiplicative digital roots are the natural numbers
, and there are no cycles other than the fixed points of
to reach a fixed point is the multiplicative persistence of
The multiplicative persistence is undefined if it never reaches a fixed point.
In base 10, it is conjectured that there is no number with a multiplicative persistence
[3][4] The smallest numbers with persistence 0, 1, ... are: The search for these numbers can be sped up by using additional properties of the decimal digits of these record-breaking numbers.
There are also additional restrictions on the first two digits.
Based on these restrictions, the number of candidates for
-digit numbers with record-breaking persistence is only proportional to the square of
However, any number that is missing from the sequence above would have multiplicative persistence > 11; such numbers are believed not to exist, and would need to have over 20,000 digits if they do exist.
[3] The multiplicative digital root can be extended to the negative integers by use of a signed-digit representation to represent each integer.
The example below implements the digit product described in the definition above to search for multiplicative digital roots and multiplicative persistences in Python.